\documentclass[11pt]{article}
\usepackage{amssymb}
\usepackage{amsmath}
%\usepackage{epsf}                                                             
%\usepackage{times} 
\usepackage{graphicx}
%\usepackage[draft]{graphicx}
%\usepackage{showkeys}
%\usepackage{refcheck}
%	page layout

\setlength{\oddsidemargin}{0.0in}
\setlength{\evensidemargin}{0.0in}
\addtolength{\textwidth}{2.5cm}
\addtolength{\textheight}{3.5cm}
\setlength{\footskip}{1.0cm}
\setlength{\topmargin}{0cm}
\setlength{\topskip}{0cm}
\setlength{\headheight}{0pt}
\setlength{\headsep}{0pt}          
\setlength{\parskip}{4pt}
\setlength{\parindent}{0pt}

\renewcommand{\baselinestretch}{1.3}

%	environments

\newtheorem{Lemma}{Lemma}[section]
\newtheorem{Theorem}[Lemma]{Theorem}
\newtheorem{Proposition}[Lemma]{Proposition}
\newtheorem{Corollary}[Lemma]{Corollary}
\newtheorem{Remark}[Lemma]{Remark}
\newtheorem{Definition}[Lemma]{Definition}
\newtheorem{Hypothesis}[Lemma]{Hypothesis}

\newenvironment{Proof}%
 {\begin{trivlist} \item[]{\bf Proof. }}%
 {\hspace*{\fill}$\rule{.3\baselineskip}{.35\baselineskip}$\end{trivlist}}

%	counters

\renewcommand{\theTheorem}{\arabic{Theorem}}
\renewcommand{\theLemma}{\arabic{section}.\arabic{Lemma}}
\renewcommand{\theRemark}{\arabic{section}.\arabic{Remark}}

\makeatletter
\@addtoreset{equation}{section}
\makeatother
\renewcommand{\theequation}{\thesection.\arabic{equation}}

%	private definitions

\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\mathr}{\mathbb{R}}
\newcommand{\MathZ}{\mathbb{Z}}
\newcommand{\mathn}{\mathbb{N}}
\newcommand{\mathc}{\mathbb{C}}

\def\Re{\mathop{\mathrm{Re}}}
\def\Im{\mathop{\mathrm{Im}}}

\newcommand{\rmO}{\mathrm{O}}
\newcommand{\rmo}{\mathrm{o}}
\newcommand{\rmd}{\mathrm{d}}
\newcommand{\rme}{\mathrm{e}}
\newcommand{\rmi}{\mathrm{i}}
\newcommand{\Rg}{\mathrm{Im}\,}
\newcommand{\Ns}{\mathrm{Ker}\,}


\newcommand{\eps}{\varepsilon}
\newcommand{\clos}{{\rm clos\,}}
\def\LL{{\cal L}}

\renewcommand{\labelenumi}{(\roman{enumi})}
\renewcommand{\theenumi}{(\roman{enumi})}

\newcommand{\reff}[1]{{\rm (\ref{#1})}}

\def\wes{\mbf w^\ast_\varepsilon}


% bold-face in mathmode

\newfam\bifam
\font\tenbi=cmmib10 scaled \magstep1
\font\sevenbi=cmmib10 at 11pt
\font\fivebi=cmmib10 at 6pt
\textfont\bifam = \tenbi
\scriptfont\bifam= \sevenbi
\scriptscriptfont\bifam= \fivebi
\def\mbf#1{{\fam\bifam\relax#1}}

\def\mbW{\mbf W}
\def\tmbW{\widetilde{\mbf W}}

%script in math-mode

\newfam\rsfsfam
\font\twlrsfs=rsfs10 scaled \magstep1 
\font\egtrsfs=rsfs7 scaled \magstep1 
\font\sixrsfs=rsfs5 scaled \magstep1 
\textfont\rsfsfam=\twlrsfs
\scriptfont\rsfsfam=\egtrsfs
\scriptscriptfont\rsfsfam=\sixrsfs
\def\scr#1{{\fam\rsfsfam\relax#1}}

\begin{document}
\begin{center}
{\Large\bf Spatio-Temporal Dynamics of Reaction-Diffusion Patterns}
\\[8mm]
{\large\em Dedicated to Willi J\"ager on the occasion of his sixtieth birthday}\\[15mm]
{\bf Bernold Fiedler$\;^1$
and
Arnd Scheel$\;^2$}
\\[20mm]
$^1${\small Freie Universit\"at Berlin, FB Mathematik I, Arnimallee
2--6, }\\
{\small 14195 Berlin, Germany}
\\[2mm]
$^2$ {\small University of Minnesota, School of Mathematics, 206 Church
St. S.E.,
}\\
{\small   Minneapolis, MN 55455, USA}
\end{center}

\newpage
\tableofcontents

\section{Introduction and overview}
%\pagestyle{plain}
%\pagenumbering{arabic}

In this survey we look at parabolic partial differential
equations from a dynamical systems point of view. With origins deeply
rooted in celestial mechanics, and many modern aspects traceable to
the monumental influence of Poincar\'e, dynamical systems theory is
mainly concerned with the global time evolution ${\cal T}(t)u_0$ of
points $u_0$ --- and of sets of such points --- in a more or less
abstract phase space $X.$ The success of dynamical concepts such as
gradient flows, invariant manifolds, ergodicity, shift dynamics,
etc. during the past century has been enormous --- both as measured by
achievement, and by vitality in terms of newly emerging questions and
long-standing open problems.

In parallel to this development, the applied horizon now reaches far
beyond the classical sources of celestial and Hamiltonian
mechanics. Applications areas today include physics, many branches of
engineering, economy models, and mathematical biology, to name just a
few. This influence can certainly be felt in several articles of this
volume and cannot possibly be adequately summarized in our survey.

Some resources on recent activities in the area of dynamics are the
book series Dynamical Systems I - X of the Encyclopedia of
Mathematical Sciences
\cite{anoarn88,sinai89,arnoetal88,arnnov90,arnold94,arnold93,arnnov94,arnold94b,anosov91,kozlov02},
the Handbook of Dynamical System \cite{broer,fiedlerhand,katok}, the Proceedings \cite{figrsp98}, and the fundamental books \cite{chowhal82,guchol83,moser73,kaha95}. 

In the context of partial differential equations, the phase space $X$
of solutions $u=u(t,x)$ becomes infinite-dimensional: typically a
Sobolev space of spatial profiles $u(t)=u(t,\cdot)$. More
specifically, the evolution of $u(t)={\cal T}(t)u_0\in X$ with time
$t$ is complemented by the behavior of the $x$-profiles $x\mapsto
u(t,x)$ of solutions $u$. Such spatial profiles could be monotone or
oscillatory; in $\dim x=1$ they could define sharp fronts or peaks
moving at constant or variable speeds, with possible collision or
mutual repulsion. Target patterns or spirals can emerge in $\dim
x=2$. Stacks of spirals, which Winfree called scroll waves, are
possible in $\dim x=3$.  For a first cursory orientation in such
phenomena and their mathematical treatment we again refer to
\cite{fiedlerhand}   and the article of Fife in the present volume, as
well as the many references there.

For our present survey, we focus on the spatio-temporal dynamics of
the following rather ``simple'', prototypical parabolic partial
differential equation:
%
\begin{eqnarray} 
u_t = D\Delta u +\; f(x,u,\nabla u).\label{1.1.1}
\end{eqnarray} 
Here $t\geq 0$ denotes time, $x\in\Omega \subseteq \mathr^m$ is space
and $u=u(t,x)\in\mathr^N$ is the solution vector. We consider
$C^1$-nonlinearities $f$ and constant, positive, diagonal diffusion
matrices $D$. This eliminates the beautiful pattern formation
processes due to chemotaxis; see \cite{jaeluc92,steven00}, for
example. Note that we will not always restrict $f$ to be a pure
reaction term, like $f=f(u)$ or $f=f(x,u)$. More general than that, we
sometimes allow for a dependence of $f$ on $\nabla u$ to include
advection effects. The domain $\Omega$ will be assumed smooth and
bounded, typically with Dirichlet or Neumann boundary conditions, or
else unbounded, typically $\Omega=\mathr^m$, an unbounded cylinder, or
a half-space. At any rate, assumptions will be such that questions
like global forward existence, uniqueness, and smoothness of weak
solutions $u(t,x)$ for prescribed initial conditions
$u_0(x)=u(t=0,x)\in X$ will not constitute a problem; see
\cite{Henry,pazy83}. To emphasize the dynamical systems aspect of
dependence on initial conditions $u_0(x)=u(t=0,x),$ we frequently
write ${\cal T}(t)u_0=u(t)=u(t,\cdot)$ for the strongly continuous
solution semigroup ${\cal T}(t)$ of (\ref{1.1.1}) on $X$.

Within this general setting, we devote the following chapters to a
discussion of increasingly specialized spatio-temporal patterns in
increasing space dimensions $m=\dim x =1,2,3$. Our approach is
motivated to some extent by classical dynamics in finite-dimensional
spaces: while flows in $\dim X=2$ admit the beautifully simple
Poincar{\'e}-Bendixson theory, present in any good textbook on
ordinary differential equations, some of the complications arising in
$\dim X=3$ are illustrated by the Lorenz attractor. Chapter \ref{c:2}
is devoted to single equations, $N=\dim u=1,$ in one space dimension,
$m=\dim x=1.$ On bounded interval domains $\Omega$, the gradient-like
dynamics is governed by a decreasing Lyapunov-functional, and by
additional nodal properties, which were first discussed by Sturm; see
Section \ref{s-2.1}. In Section \ref{s-2.2} these Sturm properties
provide a rather complete combinatorial characterization of the global
PDE attractors ${\cal A}_f$ associated to (\ref{1.1.1}). The case of a
circle domain $\Omega=S^1,$ in some sense an amphibium like $m=\dim
x=1.5,$ destroys the gradient-like Lyapunov structure, in Section
\ref{s-2.3}, but preserves the Sturm structure. We sketch a new
description of the global attractors for this case, which involves
rotating waves on the circle and their heteroclinic connections. It is
worth noting that individual orbits, but not the arbitrarily
high-dimensional global attractor ${\cal A}_f,$ do satisfy a
Poincar{\'e}-Bendixson theorem in this case. Adding a linear, but
nonlocal rank-1 perturbation can destroy this simple Morse-Smale
structure completely, allowing for embeddings of arbitrarily
complicated dynamics. Chapter~\ref{c:2} can also be read as a very
classically-minded warm-up for the much more advanced discussion of
gradient flows in the article of Fife in the present volume.

In the unbounded limit $\Omega=\mathr,$ travelling waves can arise
along with various instabilities; see Chapter \ref{s:1u}. Stability
considerations are greatly complicated --- and phenomena as well as
their mathematics greatly enriched --- by the presence of continuous
spectrum. As a prelude to our treatment of spiral wave bifurcations,
in Chapter \ref{s:4}, we discuss dispersion relations and characterize
Fredholm properties of linearizations at fronts in one space dimension
in Section \ref{s:1}. The role of group velocities is then illustrated
in a Hopf bifurcation from travelling waves caused by continuous
spectrum in Section \ref{s:1.3}. We return to bounded but large
domains $\Omega=(-L,L)$ in Section \ref{s:2}, where we investigate
linear and nonlinear stability of fronts in the limit $L\to\infty$.

Chapter \ref{s:3}, dwelling on planar domains $\Omega\subseteq
\mathr^2,$ addresses the issue of spiral waves. Following \cite{wieros46},
spiral patterns arise naturally in systems $(N=2)$ of excitable media
type. As a link to geometric dynamics and mean curvature flow, we
begin in Section \ref{s:kin} with a negative result: Archimedean
rotating spirals cannot be described by a reduced model of
curve-shortening type. Some nonlocal effects or extra input describing
the tip motion, or other relics of the system-character $N=\dim n=2$
of the viscous approximation to the geometric model, have to be
added. In Section \ref{s:3.1}, we develop a characterization of
Archimedean spiral wave solutions to reaction-diffusion systems of $N$
species, valid for excitable and oscillatory media, but independent of
singular limiting regimes such as fast relaxation kinetics or small
amplitude oscillations. Our characterization ensures robustness of
spiral waves when system parameters are varied. A bifurcation result
in Section~\ref{s:spex} provides us with an open class of reaction
diffusion systems, where spiral waves are actually proved to exist.

Beyond rigidly rotating Archimedean spiral patterns, spiral waves
generate a variety of spatio-temporal structures. In Chapter \ref{s:4}
we attempt a mathematical analysis based on bifurcation theory.  In
Section \ref{s:3.4}, we phenomenologically describe some of the
patterns that may arise, building on experimental observations. We
address in some detail meandering and drifting spirals,
Doppler-induced super-spirals, and spiral breakup in the core and
farfield. In Section \ref{s:meander} we outline a symmetry approach,
which accurately describes observed meandering cycloid bifurcations
from the unperturbed circular tip motion of rigidly rotating spiral
wave patterns. The elementary Euclidean group $SE(2)$ of rotations and
translations in the $x$-plane causes these motions by Hopf bifurcation
transverse to the group orbit: a prime example of symmetry breaking
under noncompact Lie groups.  However, this simplistic analysis
ignores the possible presence of continuous spectrum. In Section
\ref{s:3.3} we characterize spectral properties of spiral waves in
large and unbounded domains. We focus on two major consequences of the
presence of continuous spectrum in the unbounded domain. First, the
shape of eigenfunctions to isolated eigenvalues is largely governed by
the complex dispersion relation of the wavetrains in the
farfield. Second, instabilities arising through the continuous
spectrum come in two steps. At onset, the instability is convective in
nature: perturbations decay pointwise, and the instability disappears
after a possibly long transient. Only after driving the instability
further by increasing the bifurcation parameter beyond threshold, the
instability manifests itself in bounded domains. The discussion of
spectra, here, is in strong analogy to the simpler setup in one
space-dimension, Chapter~\ref{s:1u}.  We conclude Chapter~\ref{s:4}
relating spectral theory to the experimentally and numerically
observed spiral wave patterns.

Chapter \ref{c-4}, where $m=\dim x=3$, discusses scroll wave
dynamics. Following \cite{winfre73}, scroll waves can be viewed as stacks
of rotating spirals with tip positions $x=z^t(\tau)$ aligned along
filament curves $\tau\mapsto z^t(\tau)\in \Omega\subseteq \mathr^3$,
for (most) fixed times $t$. Filaments $z^t$ can be knots with
parameter $\tau\in S^1.$ Moreover the spirals, which appear in spatial
sections transverse to filaments $z^t(\tau)$ at $\tau$, define a local
phase angle $\varphi^t(\tau)\in S^1$ there. We relate knotting
orientations of filaments $z^t$, and ``twist'', alias the winding of
$\varphi^t,$ in Section \ref{s-4.1}. For generic initial conditions
$u_0(x)=u(t=0,x),$ it is possible to describe all scenarios changing
the knotting or linking topology of scroll wave filaments $z^t$; see
section \ref{s-4.2}. We conclude with a discussion of some simulations
of spatio-temporal scroll wave dynamics, in Section \ref{s-4.3}.

{\bf Acknowledgments.} We are grateful for the hospitality of the
program of the Twente Summer School and of the Seminars of DMV (German
Mathematical Society) at Oberwolfach, where these results have first
been compiled. This work was substantially supported by the Deutsche
Forschungsgemeinschaft, in particular by the Sonderforschungsbereich
"Complex Nonlinear Processes", DFG Priority Research Programs "Ergodic
Theory, Analysis, and Efficient Simulation of Dynamical Systems" and
"Analysis, Modeling, and Simulation of Multiscale Problems". For
generous support of several research visits both authors are indebted
to the Institute of Mathematics and its Applications (IMA) at
Minneapolis. The first author would like to thank Regina L\"ohr for patiently typesetting quite a few versions of this manuscript. We are also indebted to many colleagues who have helped
us along the way, most notably Sigurd Angenent, Pavol Brunovsk{\'y},
Klaus Ecker, Giorgio Fusco, Jack Hale, Jim Keener, Yannis Kevrekidis,
John Mallet-Paret, Alexander Mikhailov, Waldyr Oliva, Peter
$\rm{Pol\acute{a}\check{c}ik}$, Carlos Rocha, Bj\"orn Sandstede, Art
Winfree, and Matthias Wolfrum. 

Most of all, we feel deeply indebted to Willi J\"ager, who
untiringly created and maintained an inspiring work atmosphere at
Heidelberg with such lasting influence on his scientific children and
grandchildren.

\section{One space dimension: global attractors} 
\label{c:2}

\subsection{Lyapunov functions, comparison principles, and Sturm property}
\label{s-2.1}

In this chapter, we consider various special cases of
reaction-advection-diffusion systems of the general form
%
\begin{eqnarray} u_t &=& D\Delta u + f(x,u,\nabla u), \label{2.1.1}
\end{eqnarray} 
$\Omega\subseteq \mathr^m$, with $u\in \mathr^N,f\in C^1$, and
Dirichlet, Neumann, or mixed boundary conditions; see
(\ref{1.1.1}). Mostly we will restrict our attention to the special
case of a scalar equation, $N=1$, in one space dimension $m=1.$ We
begin our discussion, however, by recalling the larger, overlapping
domains of validity for the concepts of gradient flows or
gradient-like dynamics, as given by decreasing Lyapunov functions, and
of monotonicity or comparison principles.

\subsubsection{Lyapunov functions}
\label{s-2.1.1}
We first consider the variational structure which makes system
(\ref{2.1.1}) an $L^2$-gradient flow. We assume the vector
nonlinearity $f=f(x,u)$ to be independent of the term $\nabla u.$ In
addition, we assume $f\in\mathr^N$ to be the gradient of a scalar
potential $F=F(x,u)$ with respect to $u\in\R^N$,
%
\begin{eqnarray} F_u(x,u) &=& f(x,u). \label{2.1.2}\end{eqnarray}
Then the energy functional 
%
\begin{eqnarray} {\cal V}(u) &: =& \int_{\Omega} (\frac{1}{2}
(\sum_{j,k} \nabla u_j\cdot D_{jk} 
\nabla u_k ) -F(x,u))\, \rmd x \label{2.1.3}\end{eqnarray}
is a {\it Lyapunov function} for system (\ref{2.1.1}), if the positive
definite diffusion matrix $D$ is symmetric. Indeed we observe the
Lyapunov decay property:
%
\begin{eqnarray} 
\frac{d}{dt}{\cal V}(u(t)) &=& -\int_{\Omega} u^2_t\, \rmd x
\label{2.1.4}\end{eqnarray} 
is strictly negative, except at equilibrium solutions of the elliptic
system $u_t=0.$ Of course we have to impose appropriate growth
conditions on $f$ and choose an appropriate Sobolev space $X$
containing functions in $H^1(\Omega)$ with the chosen boundary
conditions. Critical points of the energy functional ${\cal V}$
provide weak solutions of the associated elliptic equilibrium
system $u_t=0$ and are amply studied; see for example the books
\cite{giahil96a,giahil96b,mawwil89,struwe90,zeidle85}, and the many references
there.

By LaSalles invariance principle, the monotone decay (\ref{2.1.4}) of
the Lyapunov functional ${\cal V}$ implies ${\cal V}\equiv const.$ on
the $\omega$-limit set
% 
\begin{eqnarray} 
\omega(u_0) &:=& \{ v\in X;\; v=\lim u(t_n), {\rm\;
for\;some\;}t_n\rightarrow\infty\},
\label{2.1.4b}\end{eqnarray} 
of any bounded solution $u(t)={\cal T}(t)u_0 \in X$ with initial
condition $u=u_0$ at $t=0.$

If we substitute the term $u_t$ in (\ref{2.1.1}) by $Au_t,$ with
uniformly positive definite $N \times N$-matrices $A=A(x,u,\nabla u),$
and substitute $D\Delta u$ by an $x$-dependent matrix diffusion term
${\rm div\;}(D(x){\rm \;grad\;} u)$ in divergence form, the same
functional ${\cal V}(u)$ works as a valid Lyapunov function. Indeed, the
Lyapunov decay property (\ref{2.1.4}) remains valid with $u^2_t$
replaced by the nonnegative integrand $u^T_tAu_t.$

Another generalization of (\ref{2.1.3}), due to \cite{alikik79}, relaxes
the very restrictive gradient condition (\ref{2.1.2}) under Neumann
boundary conditions and for $x$-independent $f=f(u).$ Since
homogeneous $x$-independent solutions then satisfy the ODE system
$\dot{u}=f(u),$ a gradient-like condition
% 
\begin{eqnarray} F_u (u) \cdot f(u) < 0 \label{2.1.5}\end{eqnarray}
has to be imposed, except when $f(u)=0$ provides a spatially
homogeneous equilibrium already.  An additional convexity condition
then ensures that ${\cal V}$ becomes a Lyapunov function, decreasing strictly
along any non-equilibrium solutions.

In the case of scalar equations, $N=1,$ the gradient condition
(\ref{2.1.2}) is of course trivially satisfied. Even in the
one-dimensional case $m=1,$ however, with periodic boundary conditions
$x\in S^1=\mathr/\MathZ$, Lyapunov functions may fail to exist, if we
let the nonlinearity $f=f(x,u,u_x)$ depend on the gradient term $u_x$:
% 
\begin{eqnarray} u_t := u_{xx} + f(x,u,u_x). \label{2.1.6}\end{eqnarray}
In fact a particularly simple example arises for $x$-independent
nonlinearities $f=f(u,u_x).$ Note $S^1$-{\em equivariance} with respect to
$x$-shift in that case: with $u(t,x),$ also $u(t,x-x_0)$ is a solution
of (\ref{2.1.6}), for any fixed $x$-shift by $x_0\in S^1.$ We can then
find {\it rotating wave solutions} of the form
% 
\begin{eqnarray} u(t,x) &=& U(x-ct), \label{2.1.7}\end{eqnarray}
$c\neq 0,$ which are provided by any nonstationary 1-periodic solution of the second order ODE
% 
\begin{eqnarray} U''+c U'+f(U,U')=0. \label{2.1.8}\end{eqnarray}
Such solutions $U$ are easily constructed, for suitable nonlinearities
$f$; for example the class (\ref{2.1.8}) contains the van der Pol
oscillator. See \cite{angfie88} and Section \ref{s-2.3} for more
details. Clearly the existence of rotating waves (\ref{2.1.7}) with
time period $1/c$ contradicts the existence of a Lyapunov function ${\cal V}$
which would have to strictly decrease with time. Similar constructions
work for thin annulus domains $\Omega\subset\mathr^2$ and for
disks. In fact we could just transform any system with $f=f(u)$ to
coordinates rotating at uniform angular speed $c \neq 0.$ Any
equilibrium solution which is not rotationally symmetric will then
provide a rotating wave solution for suitable $f=f(x,u,\nabla u).$

In view of such examples it is therefore perhaps surprising, and not
widely known, that a Lyapunov functional of the form
% 
\begin{eqnarray} {\cal V}(u) &=& \int_\Omega \gamma(x,u,u_x)\rmd x \label{2.1.9}\end{eqnarray}
does exist for (\ref{2.1.6}), in the case of an interval domain $\Omega
= (0,1)$ with separated boundary conditions of Dirichlet, Neumann, or
Robin type; see \cite{zeleny68,matano78}. See also the beautiful
construction in \cite{matano88}. Again the Lyapunov decay
property (\ref{2.1.4}) remains valid with $u_t$ replaced by
$Au_t$, for some uniformly positive $A=A(x,u,u_x)$.

The construction of $\gamma$ by Matano in \cite{matano88} is
particularly appealing: $\gamma$ is constructed via solutions of
subsidiary semilinear hyperbolic equations, which defy periodicity
conditions in $x$ and thus quite appropriately fail to exist under
periodic boundary conditions $\Omega=S^1$. For analogous constructions
of Lyapunov functions in the spatially discrete case of Jacobi systems
and for certain graphs of neural networks see
\cite{fieged98,fieged99}.

\subsubsection{Comparison principles}
\label{s-2.1.2}
{\it Monotonicity properties}, alias {\it comparison principles}, are
the second major structural property of reaction-advection-diffusion
equations (\ref{2.1.1}). We refer to \cite{polaci02} for a recent
survey and only sketch the main line of thought, briefly. Let
$u_0(x),\tilde{u}_0(x)$ be two initial conditions in our phase space
$X\subseteq C^0(\overline{\Omega})$, with associated (global)
solutions $u(t)={\cal T}(t)u_0,\;\tilde{u}(t)={\cal
T}(t)\tilde{u}_0\in X.$ The (strong) monotonicity property, or
parabolic comparison principle, states that
% 
\begin{eqnarray} 
u_0 \leq \tilde{u}_0 & \Rightarrow &
u(t)<\tilde{u}(t),{\rm\;for\;all\;} t>0.
\label{2.1.10}\end{eqnarray}
Here $\leq$ indicates that $u_0(x)\leq\tilde{u}_0(x),$ for all $x\in
\Omega$, whereas $<$ indicates pointwise strict ordering inside
$\Omega.$ See \cite{smolle83,walter70,prowei67} for proofs of this property in
a scalar parabolic context $N=\dim n=1,\; m=\dim x\geq 1.$ The proof
is based on the elementary observation that the difference
$w:=\tilde{u}-u$ of the two solutions satisfies a nonautonomous linear
parabolic equation of the form
% 
\begin{eqnarray} 
w_t= D\Delta w + a(t,x)w+b(t,x)\nabla w 
\label{2.1.11}\end{eqnarray}
for suitable coefficients $a,b$ which depend on the solutions
$u,\tilde{u}.$ For example
% 
\begin{eqnarray} 
a=\int^1_0 f_u(x,u+\vartheta (\tilde{u}-u),\nabla u+\vartheta(\nabla\tilde{u}-\nabla
u)) \cdot (\tilde{u}-u) \rmd\vartheta \label{2.1.12}
\end{eqnarray} where
$\tilde{u},u$ have to be evaluated at $(t,x).$ Note $w_0\geq 0.$ From
(\ref{2.1.11}) it is immediately clear that any nondegenerate local
$x$-minimum $w(t_0,x_0)=0,$ should it ever develop at some time
$t=t_0$, immediately retracts to locally strictly positive
values. Indeed (\ref{2.1.11}) then implies $w_t=\Delta w >0$ at
$(t_0,x_0).$ The rest is parabolic technique.

A very successful general framework for monotonicity properties has
been initiated by Hirsch and Matano; see
\cite{hirsch83,hirsch85,hirsch88,matano86b,matano87}. For example, Hirsch
has proved that most initial conditions, in the Baire sense, give rise
to solutions $u(t)$ which, if bounded, converge to some equilibrium in
$X$ for $t\rightarrow +\infty.$ In particular this implies that any
hyperbolic periodic solution of a (strongly) monotone system is
necessarily unstable. Let us briefly indicate a proof of this latter
fact. Indeed the compact Floquet-operator is given by $w_0\mapsto w(p)
\in X,$ with linearization along the periodic orbit of the form
(\ref{2.1.11}) and with $p$ denoting the time period. The compact
Floquet-operator therefore preserves positivity.  By Perron-Frobenius
theory, any eigenvalue of maximal modulus possesses a positive
eigenfunction. For a periodic solution $u(t)$ with trivial Floquet
multiplier 1, the trivial Floquet eigenfunction $w:=u_t$ cannot
possibly be positive. By hyperbolicity, the periodic solution
therefore possesses another Floquet multiplier outside the unit circle
and with positive eigenfunction. Instability ensues. For an excellent
survey of more recent, deeper results in this context, in particular
concerning generic convergence to {\it subharmonicity} for
time-periodically forced monotone systems, see \cite{polaci02}.

From a pure dynamics point of view, monotonicity is not a serious
restriction. As Smale has pointed out, any Lipschitz flow in
$\mathr^n$ can be embedded into $u\in\mathr^{n+1}$ --- normally
unstable, of course --- such that the resulting system is
monotone. Just view $\mathr^n$ as the hyperplane $u_1+\cdots +
u_{n+1}=0$ and add the fast scalar equation $\varepsilon\dot{v}=v$ in
the normal direction, for sufficiently small $\varepsilon >0.$ The
resulting system will be monotone with respect to the componentwise
order $\leq$ on $\mathr^{n+1}.$ Of course, this argument somewhat
neglects the spatial aspects of dynamics which are so relevant for
PDEs. Moreover, instability does not necessarily indicate invisibility
or irrelevance. Unstable solutions characterize basin boundaries of
stable objects. And sometimes their time scales of instability, which
are not at all affected by the monotonicity requirement, may
comfortably exceed the life times of their observers.

\subsubsection{Sturm property, revisited}
\label{s-2.1.3}
We now return to the scalar case, $N=\dim u=1$, in one
space-dimension, $m=\dim x=1$ with nonlinearity $f=f(x,u,u_x)$, as
specified in (\ref{2.1.6}) above. We admit separated boundary
conditions, $x\in(0,1),$ with a resulting gradient-like flow, as well
as periodic boundary boundary conditions, $x\in S^1=\mathr/\MathZ,$
which defy Lyapunov functionals and admit nontrivial time-periodic
solutions. In either case, there exists a very powerful refinement of
the above monotonicity structure: the {\it Sturm property.} For any
continuous function $w:[0,1]\rightarrow\mathr$ let the {\it zero
number} $z(w)\leq \infty$ denote the number of strict sign changes of
$w.$ In other words $z$ is the supremum of all $n$, for which we can
find $x_1<\cdots x_n$ such that the nonzero signs of $w(x_i)$
alternate. Let $w=w(t,x)$ denote a solution (\ref{2.1.11}) in one
space dimension
% 
\begin{eqnarray} w_t=w_{xx} + b(t,x)w_x + a(t,x)w. \label{2.1.13}\end{eqnarray}
Clearly $z=z(w(t,\cdot)),$ the number of sign changes of the
$x$-profiles, may then depend on $t.$ As far back as 1836, Sturm has
observed that the zero number
% 
\begin{eqnarray} z(w(t,\cdot)) \; \underset{t}{\searrow} \label{2.1.14}\end{eqnarray}
is {\it nonincreasing} with $t$, at least for time-independent
$a,b$. See \cite{stu} and also \cite{polya}.

Sturm has used his beautiful observation to prove that any nontrivial
linear combination $\varphi$ of eigenfunctions $\varphi_j,\;k\leq
j\leq \ell,$ of any Sturm-Liouville problem
% 
\begin{eqnarray}\lambda \varphi =\varphi_{xx} +b(x)\varphi_x + a(x)\varphi \label{2.1.15}\end{eqnarray}
on the interval $0\leq x\leq 1$ possesses at least $k$ and at most $\ell$ sign changes:
%
\begin{eqnarray}k \leq z(\varphi) \leq \ell. \label{2.1.16}\end{eqnarray}
Here we number the eigenfunctions $\varphi_j$ such that
$z(\varphi_j)=j.$ Defining $w(t,\cdot)$ to be the explicit exponential
solution of (\ref{2.1.13}), for all $t\in\mathr,$ with initial
condition $w=\varphi$ at $t=0$, we indeed see how $w/|w|$ approaches
eigenfunctions $\varphi_{j\pm},$ for $t\rightarrow \pm\infty.$
Invoking (\ref{2.1.14}), we obtain for some large enough $t>0$ that
%
\begin{eqnarray}\ell\geq j_- =z(w(-t,\cdot)) \geq z(w(0,\cdot)) = z (\varphi) \geq z(w(t,\cdot))=j_+ \geq
k. \label{2.1.17}\end{eqnarray}
This single line proves (\ref{2.1.16}).

It is easy to prove that (\ref{2.1.14}) indeed holds, locally, at a
nondegenerate double zero
%
\begin{eqnarray}w=w_x=0\neq w_{xx}, \quad{\rm\;at\;}(t_0,x_0).
\label{2.1.18}\end{eqnarray} 
Then (\ref{2.1.13}) implies sign $w_t={\rm\;sign\;} w_{xx},$ and an
elementary argument shows that the zero number $z(w(t,\cdot))$ drops
by 2, locally at $(t_0,x_0)\in(0, \infty)
\times[0,1].$ See also Figure \ref{Fig2.1}(a). It is less obvious that $z$ also
drops {\it strictly} at a triple, or multiple zero, where only
$w=w_x=0$ is required; see Figure \ref{Fig2.1}(b).


%
\begin{figure}[t]%Fig. 2.1
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,3.5)% 
\centerline{\includegraphics[scale=1.0]{Fig/Fig01.eps}}
\end{picture}% 
\caption{Local dropping of $z(w(t,\cdot))$ at (a) a double zero, (b) a
triple zero of 
$z(w(t_0,\cdot)).$ } \label{Fig2.1}
\end{figure}

\begin{Proposition}\label{p-2.1} Let $w(t,x)\not\equiv 0$ solve the
linear, nonautonomous equation (\ref{2.1.13}) in one space dimension
and with separated or periodic boundary conditions. Let the zero
number $z(w(t,\cdot))$ denote the number of strict sign changes of the
$x$-profiles $x\mapsto w(t,x)$, $x\in [0,1]$.

Then $w$ satisfies the {\em Sturm property,} that is
\begin{itemize}\item[(i)] $z(w(t,\cdot))$ is nonincreasing with time $t;$
\item[(ii)] $z(w(t,\cdot))$ drops strictly, whenever $w(t,\cdot)$
possesses any multiple zero  at any $x\in[0,1];$
\item[(iii)] $z(w(t,\cdot))$ is finite, for any positive $t$.
\end{itemize}
\end{Proposition} 

Note how the trivial case $z=0$ of sign definite $w$ corresponds to
the monotonicity or comparison principle (\ref{2.1.10}) discussed
above. For solutions $w(t,x)$ which are analytic in $(t,x),$ the
proposition can be proved by Newton's polygon. In the autonomous case,
this was Sturm's proof. For a modern version see \cite{angfie88}. The
definitive version of this result, under rather mild regularity
assumptions on $a,b$, was given by \cite{angene88}. We have explained
above the renewed interest in the Sturm property for {\it
nonautonomous} coefficients $a,b:$ any difference $w=\tilde{u}-u$ of
two solutions $u,\tilde{u}$ of the semilinear
reaction-advection-diffusion equation (\ref{2.1.6}) satisfies
(\ref{2.1.13}) and thus possesses the Sturm property.

We will see in the next two sections, how this simple additional Sturm
property has enormous consequences for the description of the global
spatio-temporal dynamics of (\ref{2.1.6}). It has definitely been due
to the deeply inspiring insight of Matano
\cite{matano82},
that these enormous nonlinear consequences have been first
realized. We just illustrate the beginnings of his contributions with
two further remarks here.

First, Matano's original idea was based on using the {\it lap number,}
$z(u_x(t,\cdot)),$ to characterize properties of solutions
$u(t,\cdot)$ for nonlinearities $f=f(u,u_x)$ independent of $x.$
Clearly this amounts to a choice $w=u_x$ and of course $w$ satisfies
the linearized equation (\ref{2.1.6}), which takes the form
(\ref{2.1.13}). In particular, the Sturm property implies that the
$x$-profile $u(t,\cdot)$ decomposes into finitely many intervals of
strict monotonicity, for any positive $t$.

Second, we can use the zero number $z$ and the Sturm property to prove
convergence of any bounded solution $u(t,x)$ to a single equilibrium,
for $t\rightarrow \infty,$ for $x\in [0,1]$, and under Neumann
boundary conditions. Since $w=u_t,$ like $u_x$, satisfies the
linearized equation (\ref{2.1.13}), even for nonlinearities
$f=f(x,u,u_x)$ which do depend on $x$, the Sturm property implies that
all zeros of $u_t$ must be simple, eventually, for large $t\geq t_0.$
But $u_{tx}\equiv 0$ at $x=0$, due to the Neumann boundary
condition. Therefore $u(t,x=0)$ must be eventually monotone, hence
convergent, and hence constant on the $\omega$-limit set
$\omega(u)\subseteq X.$ Therefore $u_t\equiv u_{tx}\equiv 0$ on
$\omega(u)$ and $z(u_t)$ would have to drop there, all the time. Well,
can it? No way: by the Sturm property, again, we see that $u_t\equiv
0$ on $\omega(u)$. Hence $\omega(u)$ consists entirely of equilibria
$v=v(x)$. But $v(x=0)=\lim_{t\rightarrow \infty} u(t,x=0)$ is fixed on
$\omega(u)$. Together with the Neumann boundary condition
$v_x(x=0)=0,$ this specifies $v\in\omega(u)$ uniquely: the
$\omega$-limit set always consists of just a {\it single}
equilibrium. Note how this argument provides a second proof of the
gradient-like behavior of the scalar, one-dimensional
reaction-advection-diffusion equation (\ref{2.1.6}), which is
completely independent of the approach by Zelenyak and Matano
\cite{zeleny68,matano78,matano88} via Lyapunov functions
(\ref{2.1.9}).


\subsection{Sturm attractors on the interval} 
\label{s-2.2}
In this section and the following one we describe some progress during
the last 15 years in our understanding of global attractors of the
scalar, one-dimensional reaction-advection-diffusion equation
% 
\begin{eqnarray}
u_t &=& u_{xx} + f(x,u,u_x). 
\label{2.2.1}\end{eqnarray}
The present section addresses the interval case $0<x<1$ under separated boundary conditions;
to be completely specific we consider Neumann boundary 
% 
\begin{eqnarray}u_x = 0 \quad {\rm\;at\;}\quad x=0,1. \label{2.2.2}\end{eqnarray}
For periodic boundary conditions $x\in S^1=\mathr/\MathZ$ and
nonlinearities $f=f(u,u_x)$ see Section \ref{s-2.3}.

\subsubsection{Global attractors}
\label{s-2.2.1}
For an excellent recent survey on the theory of global attractors,
including topics like dimension estimates, inertial manifolds,
determining modes and applications to retarded functional differential
equations, damped hyperbolic wave equations, and the gradient-like
parabolic equations studied here, we refer to the book by Chepyzhov and Vishik 
\cite{chevis02}, which includes the nonautonomous case, and to the
article by Raugel \cite{raugel02}. For earlier work on global
attractors see for example the books
\cite{hale88,temam88,ladyzh91,babvis92} and the many references there.

Abstractly, the concept is the following. Recall that ${\cal T}(t)u_0:= u(t)$
denotes the solution semigroup of (\ref{2.2.1}) on a suitable Banach
space $X$ of $x$-profiles $u(t)=u(t,\cdot)\in X.$ We require $f\in
C^2$ to be {\it point dissipative,} that is, there exists a (large)
ball in $X$, in which any solution ${\cal T}(t)u_0$ stays eventually, for all
$t\geq t_0 =t_0(u_0).$ Explicit sufficient conditions on $f$ require,
for example,
% 
\begin{eqnarray}f(x,u,0) \cdot u < 0 \label{2.2.3}\end{eqnarray}
for all sufficiently large $|u|$, along with uniformly subquadratic
growth of $f$ in the variable $u_x$. Condition (\ref{2.2.3}) provides
an $L^\infty$-bound for $u$, whereas subquadratic growth prevents
blow-up of $u_x.$ Since ${\cal T}(t)$ is a compact, point dissipative
semigroup, in the sense of the references quoted above, the {\em
global attractor} ${\cal A} = {\cal A}_f$ of (\ref{2.2.1}) can be
equivalently characterized as follows:
% 
\begin{eqnarray}\begin{array}{rcl} \cal{A} & = & {\rm the\;smallest\;
set\; attracting\; all\; bounded\; 
sets\;}\\ &=& {\rm the\;largest\; compact\; invariant\; set\;}\\ &=&
{\rm the\; set\; of\; all\; bounded\; solutions\;} {\cal T}(t)u_0 \in
X,\;t\in\mathr
\end{array}\label{2.2.4}\end{eqnarray} 
Here {\it attractivity} is understood in the sense that ${\cal T}(t){\cal B}$
stays in any arbitrarily small $\delta$-neighborhood of $\cal{A},$ for
any bounded set ${\cal B}\subset X$ of initial conditions and for all
$t\geq t_0({\cal B},\delta).$ {\it Invariance} is understood in both
forward and backward time. Negative time invariance of $\cal{A}$
requires the existence of a {\it past history} $u(-t)\in {\cal A},\;
t\geq 0,$ of $u_0,$ such that ${\cal T}(t)u(-t)=u_0$ for all $t>0.$
Similarly, the set of all bounded solutions is understood to consist
of precisely those $u_0$ which possess a bounded past history, in
addition to ${\cal T}(t)u_0$ being bounded uniformly for $t\geq 0.$ We need
not really bother about the mind-boggle of huge fans of past histories
of $u_0$ here, because backwards uniqueness for parabolic equations
implies that past histories are uniquely determined.

Under suitable dissipativeness conditions on the nonlinearity $f$, the
above statements hold true in complete generality (\ref{1.1.1}),
including systems $u\in \mathr^N,N\geq 1$, higher-dimensional
smoothly bounded $\Omega\subseteq \mathr^m,m\geq 1$, or the circle case
$\Omega=S^1.$ The gradient-flow property of (\ref{2.2.1}) with Neumann
boundary conditions (\ref{2.2.2}), however, allows for an even more
specific characterization of the global attractor:
% 
\begin{eqnarray} 
\cal{A} & = & \cal{E} \cup {\rm \{heteroclinics\} }.
\label{2.2.5}\end{eqnarray} 
Here ${\cal E}=\{u_t=0\}\subset X$ denotes the set of equilibria. We
call $u_0\in X$ {\it heteroclinic}, if there exists a solution
$u(t),t\in\mathr,$ including a past history, such that the distinct
$\alpha$- and $\omega$-limit sets of $u_0$ for $t\rightarrow
\pm\infty$ are both equilibria. We have already seen in section 2.1
how these sets will consist of a single equilibrium $U_\pm$,
respectively, as $t\rightarrow \pm\infty.$ The Lyapunov function
${\cal V}$, which only exists under separated boundary conditions,
implies ${\cal V}(U_+) < {\cal V}(U_-)$ and hence $U_+\neq U_-$:
heteroclinicity. In fact {\em homoclinic} solutions with $U_+\equiv
U_- \not\equiv u(t)$ are possible for $x\in S^1;$ see \cite{sanfie92}
and the remarks at the end of Section \ref{s-2.3}.

\subsubsection{Sturm attractors and Sturm permutations}
\label{s-2.2.2}
For simplicity of presentation, we henceforth assume that all
equilibria $U\in {\cal E}$ are {\it hyperbolic}: all eigenvalues
$\lambda$ of the Sturm-Liouville eigenvalue problem (\ref{2.1.15}),
arising by linearization at $U$, are nonzero. Such an assumption holds
for most nonlinearities $f$, in a Baire sense: zero has to be a
regular value of $u\mapsto u_{xx}+f.$ Hyperbolicity implies that
equilibria $U$ are locally isolated and, by compactness of ${\cal
E}\subseteq {\cal A},$ finite in number,
% 
\begin{eqnarray}\cal{E} & = & \{ U_1,\cdots, U_n
\}. \label{2.2.6}\end{eqnarray} Moreover, they are accompanied by a
local saddle structure, much in the geometric spirit of ODEs,
characterized by local stable and unstable manifolds $W^s_j,W^u_j$ at
each equilibrium $U_j.$ These invariant manifolds consist of those
$u_0\in X$ which remain in a neighborhood of $U_j$ and converge to
that same equilibrium in forward resp. backward time. Their tangent
spaces at $U_j$ are of course spanned by the Sturm-Liouville
eigenfunctions $\varphi$ of (\ref{2.1.15}) with negative
resp. positive eigenvalues $\lambda.$ The unstable manifolds $W^u_j$
in fact extend forward, globally, by the solution semigroup ${\cal
T}(t), t\geq 0.$ Noting that trivially $U_j\in W^u_j,$ even if $U_j$
is stable, the above characterization (\ref{2.2.5}) of the global
attractor therefore takes the equivalent form
% 
\begin{eqnarray}
\cal{A} & = & \displaystyle{\bigcup^n_{j=1}}\;
W^u_j. 
\label{2.2.7}\end{eqnarray}  


So much for generalities. As to the specific determination of global
attractors ${\cal A}_f$ associated to nonlinearities $f$ of the
parabolic equation (\ref{2.2.1}), and in fact for any gradient-like
system, several basic questions arise:
%
\begin{itemize}
\item[{\bf Q0}:] What are the equilibria $U_1,\cdots, U_n$?
\item[{\bf Q1}:] Given all equilibria, which pairs possess a
heteroclinic orbit?
\item[{\bf Q2}:] Given dissipative nonlinearities $f,\tilde{f}$, how
can we determine whether or not their global attractors ${\cal A}_{f},
{\cal A}_{\tilde{f}}$ coincide?
\item[{\bf Q3}:] How many different global attractors with $n$
equilibria are there?
\item[{\bf Q4}:] What is the geometry of the global attractors?
\end{itemize} 

Note that {\bf Q0} is an ODE question concerning the scalar second
order equation
% 
\begin{eqnarray}0 & = & U_{xx}+ f(x,U,U_x) \label{2.2.8}\end{eqnarray}
with Neumann boundary conditions. See \cite{schaaf90} for a detailed
analysis of the case $f=f(U)$. In contrast, question {\bf Q1} --
{\bf Q4} address the PDE (\ref{2.1.1}). We will now summarize some
results which show how some essentially combinatorial information on the
ODE question {\bf Q0} is sufficient to answer the PDE questions {\bf
Q1} and {\bf Q2}. In particular we will make precise what we mean by
``coinciding'' attractors. The challenging questions {\bf Q3}, {\bf
Q4}, must remain open at this time.

Since our answers to {\bf Q0} -- {\bf Q2} for the parabolic equation
(\ref{2.1.1}) are crucially determined by the Sturm property of
solutions, Proposition \ref{p-2.1}, we call the associated global
attractors {\it Sturm attractors.}

The combinatorial ODE information on the equilibria $U_1,\cdots,U_n$
has been ingeniously distilled by \cite{fusroc91}. Under Neumann
boundary conditions, let the equilibria $U_1,\cdots,U_n$ be numbered
according to the strict ordering
% 
\begin{eqnarray}U_1< U_2 < \cdots < U_n,\quad {\rm at\;}
x=0. \label{2.2.9}\end{eqnarray} Indeed $U_x=0$ at $x=0,$ together
with the second order type of the equilibrium equation (\ref{2.2.8}),
implies that $i=j,$ if $U_i=U_j$ at $x=0$. Now define the {\it Sturm
permutation} $\pi=\pi_f\in S_n$ by the corresponding, possibly
different ordering at the other boundary point $x=1$:
% 
\begin{eqnarray}U_{\pi(1)} < U_{\pi(2)} < \cdots U_{\pi(n)},\quad {\rm
at\;} x=1 \label{2.2.10}\end{eqnarray}  
To get used to Sturm permutations, let us quickly prove that
$\pi(1)=1$,\ \ $\pi(n)=n.$ Indeed choose a very large positive or negative
initial condition $u_0^\pm:$ pointwise above resp. below any
equilibrium $U_j.$ By dissipativeness, the solutions ${\cal
T}(t)u_0^\pm$ must then each converge to some equilibrium $U^\pm.$ By
monotonicity (\ref{2.1.10}), alias nonincrease of $z(u(t)-U_j)=0,$
that equilibrium must still lie above, resp. below, all other
equilibria, pointwise. Hence $U^+=U_n$ and $U^-=U_1.$ This proves
$\pi(1)=1$,\ \ $\pi(n)=n$.

Question {\bf Q2} was answered by \cite{fieroc00} as follows.

\begin{Theorem}\label{t-2.2.1} Let $f,\tilde{f}$ be dissipative
nonlinearities in (\ref{2.2.1}), (\ref{2.2.2}) such that their Sturm
permutations $\pi_f$ and $\pi_{\tilde f}$ coincide. Then the
associated global attractors, the {\em Sturm attractors} ${\cal A}_f$
and ${\cal A}_{\tilde f},$ are $C^0$ orbit equivalent. In symbols
% 
\begin{eqnarray}\pi_f = \pi_{\tilde f}\quad \Rightarrow \quad {\cal
A}_f \cong {\cal A}_{\tilde f} 
\label{2.2.11}\end{eqnarray}  
\end{Theorem}

We recall that $C^0$ orbit equivalence is given by a homeomorphism
$H:{\cal A}_f \rightarrow {\cal A}_{\tilde f},$ in the topology of
$X$, which maps $f$-orbits in ${\cal A}_f$ onto $\tilde{f}$-orbits in
${\cal A}_{\tilde{f}},$ preserving time direction.

We sketch some of the ingredients to the proof of this theorem. Of
central importance in the {\it Morse-Smale property} on attractors
${\cal A}$ with hyperbolic equilibria: all intersections $W^u_i \cap
W^s_j$ of stable and unstable manifolds are transverse, automatically,
due to the Sturm property of the semiflow. This striking fact was
discovered independently by \cite{angene86b} and \cite{henry85a}. An
indirect zero number argument runs as follows. Consider a heteroclinic
orbit $u(t)\in W^s_k$ from $U_j$ to $U_k$. If transversality failed,
we could choose a nontrivial solution $\psi(t)$ of the adjoint
linearization, along $u(t),$ perpendicular to the tangent spaces
% 
\begin{eqnarray}T_{u(t)} &:= & T_{u(t)} W^u_j +
T_{u(t)}W^s_k. \label{2.2.12}\end{eqnarray}  
Letting $t\rightarrow -\infty,$ we then see
% 
\begin{eqnarray} u(\psi(t)) \geq i_j,\quad {\rm
for\;all\;}t. \label{2.2.13}\end{eqnarray}  

Here $i_{j}$ denotes the unstable dimension, alias the {\em Morse
index} at the hyperbolic equilibrium $U_j.$ Indeed the adjoint
linearization is parabolic in backwards time and hence $z(\psi(t))$
increases with $t.$ Sturm-Liouville theory then shows the inequality
(\ref{2.2.13}) ``at'' $t=-\infty,$ because $\psi(t)\perp T_{u(t)}
W^u_j.$ A similar argument at $t=+\infty$ proves
% 
\begin{eqnarray}z(\psi(t)) < i_k, \quad {\rm\;for\;
all\;}t. \label{2.2.14}
\end{eqnarray} 
Together, this implies $i_j<i_k$ for the Morse indices of the source
$U_j$ and the target $U_k.$ Arguing for the heteroclinic tangent $u_t$
itself, on the other hand, and using that $u_t/|u_t|$ converges to
eigenfunctions for $t\rightarrow \pm \infty,$ by \cite{brufie86a}, we
see that the Sturm property for the linearization along $u(t)$ implies
% 
\begin{eqnarray}
i_k\leq z(u_t(t)) < i_j, \quad {\rm\;for\;all\;}t. \label{2.2.15}
\end{eqnarray}  
This is a clear contradiction to $i_j < i_k$, and hence proves the
Morse-Smale property.

For dynamical systems on compact manifolds, Palis and Smale have
proved {\em structural stability} of Morse-Smale systems; see
\cite{palis69,palsma70}. Infinite-dimensi\-onal versions are due to
Oliva; see \cite{oliva02}. In our setting this implies the claim of Theorem
\ref{t-2.2.1}, provided that $\tilde{f}$ is near $f$ in the
$C^2$-topology. It also proves the claim, if we can find a homotopy
$f^\tau$ from $f^0=f$ to $f^1=\tilde{f}$, in the class of dissipative
$C^2$-nonlinearities and parabolic equations (\ref{2.2.1}), such that
all equilibria in ${\cal E}_\tau= {\cal E}_{f^\tau}$ remain hyperbolic
throughout this homotopy. Unfortunately, however, it is still not
clear to us whether or not such a homotopy $f^\tau$ exists.

We have circumvented this difficulty by discretization within the
class of {\em Jacobi systems}, also studied by Oliva, see
\cite{oliva02}. These are finite tridiagonal nonlinear systems
% 
\begin{eqnarray}\dot{u}_i &=& f_i(u_{i-1},u_i,u_{i+1}),
\label{2.2.16}\end{eqnarray}  
$i=1,\cdots,\ell,$ such that the off-diagonal partial derivatives of
all $f_i$ are strictly positive everywhere. Boundary conditions like
$u_0:=u_1,\;u_{\ell+1} :=u_\ell$ have to be imposed. Counting strict
sign changes along the discrete positions $1\leq i\leq \ell,$ instead
of $0\leq x\leq 1,$ provides a Sturm property which is completely
analogous to the continuous parabolic case. The Morse-Smale property
follows, likewise, from hyperbolicity of equilibria.  Finite
difference discretization of (\ref{2.2.1}) for $f$ and $\tilde{f}$,
for example, leads to Jacobi systems $\ell$ large enough to ensure
structural stability of the respective finite-dimensional global
attractors under discretization.

But not all Jacobi systems are discretizations of parabolic
equations. The additional freedom thus gained, adorned with an
artificial unstable suspension trick which quadruples the dimension of
the system, is sufficient to find a homotopy $f^\tau$ as described
above, but on the discrete level. This completes our sketch of the
proof of Theorem \ref{t-2.2.1}.

As an aside we note that the class ${\texttt A}^J$ of (Morse-Smale)
Sturm attractors of Jacobi systems coincides with the same class
${\texttt A}^P$ for one-dimensional parabolic
equations. Discretization shows ${\texttt A}^J\subseteq {\texttt
A}^P.$ The combinatorial characterization of possible Sturm
permutations, given in Theorem \ref{t-2.4} below, shows that ${\texttt
A}^J \subseteq {\texttt A}^P;$ see \cite{fieroc00}. Similarly, the
class ${\texttt A}^P$ is independent of the choice of separated
boundary conditions; see \cite{Fie7}. Moreover, this suggests
extensions to quasilinear or nonlinear parabolic equations: as long as
finite-difference discretization to Jacobi systems prevails, no new
attractors are to be expected. An open question, however, is
realization of Sturm attractors ${\cal A}$ by Jacobi systems of
minimal dimension. Let
% 
\begin{eqnarray}\ell := \dim {\cal A} = \max_{1 \leq j \leq n} i_j
\label{2.2.17}\end{eqnarray}  
denote the maximal Morse-index of equilibria $U_j\in {\cal A}.$ Is it
then possible to realize that same attractor ${\cal A}$, up to $C^0$
orbit equivalence, by a Jacobi system (\ref{2.2.16}) of dimension only
$\ell$? A positive answer would provide the ultimate ``qualitative''
discretization.

Similar realization questions in the class ${\texttt A}^P$ of Sturm
attractors immediately arise when we restrict the class of admissible
nonlinearities $f=f(x,u,u_x).$ For example it seems possible to
exhaust the class ${\texttt A}^P$ of Sturm attractors by ``Hamiltonian''
nonlinearities $f=f(x,u)$ alone \cite{wolper}. On the other
hand, consider nonlinearities with $f(-x,u,-p)=f(x,u,p),$ for all
$x,u,p.$ Then the reflection symmetry $x\mapsto 1-x$ immediately
implies that $\pi_f$ is an involution: $\pi_f=\pi_f^{-1}$. In
particular, nonlinearities $f=f(u)$ which depend neither on $x$ nor on
$u_x$, explicitly, do not generate the class ${\texttt A}^P$
completely. Indeed there exist Sturm permutations $\pi_f$ with $n=9$
equilibria, which contain 3-cycles and hence fail to be involutions;
see Figure \ref{Fig2.2} below. The minimal such example requires $n=7$
equilibria and is given by $\pi_f=(2,4,6)(3,5)$.

\subsubsection{Sturm permutations and heteroclinics}
\label{s-2.2.3}
We address question {\bf Q1} next: how to determine all heteroclinics
between equilibria? A first answer is given by the following theorem
of \cite{fieconstr,wolfconstr}.
\begin{Theorem}\label{t-2.3}
 The Sturm permutation $\pi=\pi_f,$ defined by (\ref{2.2.9}),
(\ref{2.2.10}) above, determines explicitly and constructively for all
equilibria $U_j,U_k,\;1\leq j,k \leq n$ of (\ref{2.2.1}),
(\ref{2.2.2})
\begin{itemize} 
\item[(i)] the Morse indices (unstable dimensions) $i_j,i_k$ of
$U_j,U_k$
\item[(ii)] the zero numbers $z(U_k-U_j)$
\item[(iii)] whether or not there exists a heteroclinic solution
$u(t)$ from $U_j$ to $U_k.$
\end{itemize}
\end{Theorem}


The ``explicit and constructive'' algorithm asserted by the theorem is
slightly involved. For example
% 
\begin{eqnarray}i_k\;=\;\sum^{k-1}_{j=1}\; (-1)^{j+1} {\rm \;sign\;}
(\pi^{-1}(j+1)-\pi^{-1}(j)), 
\label{2.2.18}\end{eqnarray}   
and a similar formula holds for the zero numbers $z(U_k-U_j);$ see
\cite{fieconstr}. Note that indeed $i_1=0$ indicates stability of the
minimal equilibrium $U^-$ as was to be expected from dissipativeness
and the monotone convergence to $U^-=U_1$ mentioned above. Similarly,
$i_n=0.$ In particular, this implies that the total number $n$ of
equilibria is odd: just regard the $k-1$ entries $\pm 1$ in the sum
(\ref{2.2.18}) which must add up to zero, for $k=n.$ Alternatively,
this oddness can easily be derived from Leray-Schauder degree theory
of the equilibria, or from the classical Morse inequalities.

As for assertion (iii) of Theorem \ref{t-2.3}, we mention a {\em
cascading principle} which is peculiar to Sturm attractors. Whenever
there is a heteroclinic orbit from $U_j$ to $U_k$, there exists a
cascade of heteroclinic orbits from $U_{j_0}=U_j$ through
$U_{j_1},\cdots, U_{j_{\ell-1}}$ to $U_{j_\ell}=U_k$ such that the
Morse indices drop by $1$ along each heteroclinic in the cascade:
% 
\begin{eqnarray}
i_{j_\ell} &=& i_{j_{\ell+1}} +1. 
\label{2.2.19}\end{eqnarray} 
By transversality, each heteroclinic orbit in the cascade is just an
isolated orbit, locally. It is in fact unique. Already the simple
example of a gradient flow with respect to the height function on the
standard Euclidean 2-sphere shows that the cascading principle fails for
Morse-Smale flows, in general.

{\em Transitivity,} in contrast, is a general principle for
Morse-Smale flows: if there are heteroclinics from $U_{j_1}$ to
$U_{j_2}$ and from $U_{j_2}$ to $U_{j_3}$, then there also exists a
direct heteroclinic from $U_{j_1}$ to $U_{j_3}.$ The proof is based on
the $\lambda$-Lemma.

Combined, cascading and transitivity reduce the problem of finding
heteroclinics from $U_j$ to $U_k$ to the case $i_j=i_k+1$. Here the
zero number and the Sturm property strike again.  Such a heteroclinic
exists if, and only if, there does not exist an equilibrium $U,$
between $U_j$ and $U_k$ at $x=0,$ such that
% 
\begin{eqnarray}z(U_j-U) & \leq &
z(U_k-U). \label{2.2.20}
\end{eqnarray} 
The ``only if'' part indeed follows indirectly from Proposition
\ref{p-2.1} Just consider
% 
\begin{eqnarray}w(t) & := & u(t)-U \label{2.2.21}\end{eqnarray}  
along a hypothetical heteroclinic $u(t)$ from $U_j$ to $U_k.$ Since
$U$ lies between $U_j$ and $U_k$, at $x=0,$ and since $w_x \equiv 0$
there by the Neumann boundary conditions for $u(t)$ and $U$, a
multiple zero of $w$ has to arise at $x=0,$ for some $t=t_0.$
Therefore $z(w(t))$ must drop strictly at least once, in contradiction
to (\ref{2.2.20}). Thus $U$ as in (\ref{2.2.20}) {\em blocks} any
heteroclinic from $U_j$ to $U_k.$

The ``if'' part is proved by a Conley index argument and by a suitable
homotopy to a saddle-node bifurcation situation for $U_j$ and
$U_k$. By the explicit formulae for the Morse indices and zero numbers
of parts (i) and (ii) of Theorem \ref{t-2.3}, this procedure enables us to
recursively determine all heteroclinics in the Sturm attractor ${\cal
A}_f$ from the combinatorial ODE information on the Sturm permutation
$\pi=\pi_f$ alone.

To illustrate Theorems \ref{t-2.2.1}, \ref{t-2.3} we ask for the Sturm
attractor ${\cal A}$ of {\em maximal dimension} $\ell=\max_k i_k,$
with a fixed (odd) number $n$ of equilibria. By our explicit
expression (\ref{2.2.18}) for the Morse indices $i_k$ in terms of the
Sturm permutation $\pi,$ we first observe $i_{k+1}=i_k\pm 1.$ Since
$i_0=i_n=0$ are stable, this implies a bound $2\ell+1 \leq n.$ Indeed
$i_k$ must ascend from $i_0=0$ to the maximal value $\ell$, and also
descend back to $i_n=0.$ A combinatorial exercise then shows that the
maximality requirement $2\ell+1=n$ for $\ell$ is satisfied if, and
only if, the permutation $\pi$ is given by
% 
\begin{eqnarray}\pi=\pi_{CI} & := & (2,\; n-1)(4,\; n-3)\cdots
(\ell',\;n+1-\ell') \label{2.2.21b}\end{eqnarray}  
where $\ell'$ denotes the largest even integer not exceeding
$\ell=(n-1)/2.$ By elementary ODE phase plane methods (``time map'',
see for example \cite{chainf74,smolle83,brucho84,schaaf90,birrot62},
the permutation $\pi_{CI}$ arises in the {\em Chafee-Infante problem}
% 
\begin{eqnarray}f & = & \mu u(1-u^2). \label{2.2.21c}\end{eqnarray} 
Indeed $\pi_f=\pi_{CI}$ for $\lambda_{\ell-1}<\mu < \lambda_\ell,$ where
$\lambda_k:= (k\pi)^2$ and $\ell=\dim {\cal A}_{CI}$ is the Morse-index of
the trivial equilibrium $U\equiv 0.$ The maximal Morse index $\ell$ is
indeed attained at $U\equiv 0,$ only. By Theorem {t-2.2.1}, we can now
conclude that {\em the} Sturm attractor ${\cal A}$ of maximal
dimension $\ell$ coincides with the {\em Chafee-Infante attractor}
${\cal A}_{CI}:={\cal A}_f,$ for $f$ as in (\ref{2.2.21c}), up to
$C^0$ orbit equivalence. For an interesting geometric description of
the Chafee-Infante attractor see \cite{hatmis91}.

Wolfrum has developed a much more elegant approach to Theorem
\ref{t-2.3}, based on his notions of {\em $z$-ordering} and {\em
$z$-adjacency}. For $z=0,1,2,\cdots$ define partial orders $<_z$ on
the equilibria ${\cal E} =\{U_1,\cdots,U_n\}$ as follows: $U_j<_z U_k$
if, and only if,
% 
\begin{eqnarray}
z(U_j-U_k)=z \quad{\rm\;and\;}\quad U_j<U_k\quad{\rm\;at\;}\quad x=0.
\label{2.2.22}\end{eqnarray}  
Clearly heteroclinics between such $z$-ordered $U_j,U_k$ are blocked
in the sense of (\ref{2.2.20}), by the Sturm property, unless $U_j$
and $U_k$ are adjacent in this order. Of course $z$-adjacency means
that there does not exist another equilibrium $U$ such that $U_j<_z
U<_z U_k.$ We caution the reader that the $z$-order $<_z$ is not, in
general, transitive.

Wolfrum has proved the following; see \cite{wolfconstr}.

\begin{Theorem}\label{t-2.4}
Let $U_j,U_k$ be distinct equilibria with Morse indices $i_j,i_k$ in
the Sturm attractor ${\cal A}_f$ of (\ref{2.2.1}), (\ref{2.2.2}) on
the interval. Let $z:=z(U_j-U_k)$. Then there exists a heteroclinic
orbit from $U_j$ to $U_k$ if, and only if, $i_j>i_k$ and $U_j,U_k$ are
$z$-adjacent.
\end{Theorem}

This very concise and direct description of all heteroclinic orbits is
again purely combinatorial in terms of the Sturm permutation
$\pi=\pi_f.$ The necessary data $i_j,i_k,z(U_j-U_k)$ are provided by
the explicit formulae of parts (i), (ii) of theorem 2.2; see for
example (\ref{2.2.18}). The result was not, however, derived from that
theorem. Instead of cascading, transitivity, and blocking, it is based
on a refined geometric analysis of the transversality and zero-number
properties along the hierarchies of strong unstable and stable
manifolds of all equilibria $U_j.$ These strong manifolds are
submanifolds of $W^u_j,W^s_j$ characterized by the faster exponential
convergence rates associated to the higher unstable and stable
eigenvalues; see for example \cite{brufie86a}.

\subsubsection{Combinatorics of Sturm attractors}
\label{s-2.2.4}

Theorems \ref{t-2.2.1} -- \ref{t-2.4} highlight the central importance
of the Sturm permutation $\pi_f$ for the characterization of Sturm
attractors ${\cal A}_f.$ It is therefore worthwhile to note that there
exists a purely combinatorial characterization of the Sturm permutations,
which does not recur to any specific dissipative nonlinearities
$f=f(x,u,u_x)$. See \cite{fieroc99} for the following result.

\begin{Theorem}\label{t-2.5} 
Let $\pi\in S_n$ be any permutation.  Then $\pi$ is a Sturm
permutation, that is, $\pi=\pi_f$ for some dissipative nonlinearity
$f=f(x,u,u_x)$ in (\ref{2.2.1}), if, and only if, $n$ is odd and $\pi$
is a {\em dissipative Morse meander.}
\end{Theorem}

Here a permutation $\pi\in S_n$ is called {\em dissipative}, if
$\pi(1)=1$ and $\pi(n)=n.$ The permutation $\pi$ is called {\em
Morse}, if $\iota_k\geq 0,$ holds for all {\em Morse numbers}
% 
\begin{eqnarray}\iota_k := \sum^{k-1}_{j=1}(-1)^{j+1} {\rm\;sign\;}
(\pi^{-1}(j+1) -\pi^{-1}(j)), 
\label{2.2.23}\end{eqnarray}  
$k=1,\cdots,n.$ Of course, $\iota_1:=0$ anyways. To define the {\em
meander} property of a permutation choose any two distinct numbers
$1\leq j, k\leq n$ of the same even/odd parity, such that
$\pi^{-1}(k)$ is between $\pi^{-1}(j)$ and $\pi^{-1}(j+1)$. Then $\pi$
is called a meander permutation, if necessarily $\pi^{-1}(k+1)$ is
also between $\pi^{-1}(j)$ and $\pi^{-1}(j+1)$, for any such choice of
$j,k.$

We briefly indicate why Sturm permutations $\pi=\pi_f$ are necessarily
dissipative Morse meanders. The converse direction, which is much more
involved, requires the construction of a nonlinearity $f=f(x,u,u_x)$
realizing a prescribed permutation $\pi$ as its Sturm permutation, and
will be omitted.

Immediately following our definition (\ref{2.2.9}), (\ref{2.2.10}) of
Sturm permutations $\pi_f$, we have already indicated how the
monotonicity of the semiflow implies that $\pi_f$ is dissipative. The
Morse property follows because the Morse numbers $\iota_k$ defined in
(\ref{2.2.23}) coincide with the Morse indices $i_k$ of the equilibria
$U_k$ for Sturm permutations $\pi=\pi_f$; see (\ref{2.2.18}) and
\cite{ro85,ro91,fusroc91}. Of course $i_k\geq 0$ holds for the Morse indices,
which are the dimensions of the unstable manifolds $W^u_k.$

To understand the meander property we recall Arnold's definition
\cite{arnvis89} of a {\em meander permutation}. Consider a connected oriented
non-selfintersecting curve in the plane, intersecting a fixed oriented
base line in $n$ points. The intersections are assumed to be strict
crossings. The permutation defined by ordering the intersection
points, first along the base line and then along the curve, is called
a meander permutation. Here we label the intersection points by
$1,\cdots,n,$ when ordered along the base line, and by
$\pi^{-1}(1),\cdots,\pi^{-1}(n)$ when ordered along the curve. This
geometric definition of a meander permutation $\pi$ indeed coincides
with ours, by the Jordan curve theorem.

We now show that Sturm permutations $\pi=\pi_f$ indeed possess the
meander property. We first solve the Neumann boundary-value problem
% 
\begin{eqnarray}
0 := U_{xx}+f(x,U,U_x) 
\label{2.2.24}\end{eqnarray} 
for equilibria $U=U(x)$ by a ``shooting method'' in the phase plane
$(U,U_x).$ Consider the base line of initial conditions
$(U,U_x)=(a,0),a\in\mathr$, at $x=0$, given by the $U$-axis.  Solving
the initial value problem (\ref{2.2.24}) up to $x=1,$ for this line of
initial conditions, we obtain a differentiable curve at $x=1$ in the
$(U,U_x)$-plane, which we call the {\em shooting curve}. The shooting
curve is parameterized by $a.$ Intersection points of the shooting
curve with the base line $U_x=0$ are equilibria: they correspond to
the solutions $U_1,\cdots, U_n$ of the Neumann problem. The labeling
by $1,\cdots,n$ along the base line corresponds to the ordering of
these solutions at $x=1.$ The labeling by
$\pi^{-1}(1),\cdots,\pi^{-1}(n)$ along the shooting curve corresponds
to the ordering by $a$ at $x=0.$ The shooting curve is a connected
Jordan curve because the ODE-flow (\ref{2.2.24}) from $x=0$ to $x=1$
defines a diffeomorphism of the $(U,U_x)$-plane. Hyperbolicity of the
equilibria $U_1,\cdots,U_n$ corresponds to transverse intersections of
the shooting curve with the base line $U_x=0;$ see
\cite{brucho84,ro85}. These observations clearly prove that Sturm
permutations are indeed dissipative Morse meanders.

With the help of our combinatorial characterization of Sturm
permutations it is now possible to characterize Sturm attractors. See
Table \ref{ta:1}  for numbers of Sturm permutations $\pi\in S_n,
n=1,\cdots,17.$ We note that the trivial transformation $x\mapsto -x$
changes Sturm permutations $\pi$ to their inverse $\pi^{-1}.$
Similarly $u\mapsto -u$ conjugates $\pi\in S_n$ with the involution
$j\mapsto n+1-j$ in $S_n$. Neither symmetry operation changes the
Sturm attractor, geometrically. See Figure \ref{Fig2.2} for all
sixteen Sturm attractors with 9 equilibria  together
with their Sturm permutations, reduced by the above trivial
symmetries \cite{fiedle94}.

Observe that apparently unrelated Sturm permutations, of different
cycle lengths and hence non-conjugate, may lead to the ``same'' Sturm
attractor. Interestingly, these cases differ by the geometry of fast
unstable manifolds, as was discovered by Wolfrum \cite{woldis}; see
Figure \ref{Fig2.3}. In general, it is still an open question whether
this geometric distinction is sufficient to provide a 1-1
correspondence between Sturm attractors ${\cal A}_f$ and Sturm
permutations $\pi_f,$ up to the above trivial symmetries.


%%%%%
\begin{table} \centering
\begin{tabular}{|r|c|c|c|}%[h] 
\hline n & dissipative meanders & Sturm permutations & Sturm mod
symmetry \\ \hline \hline 1 & 1 & 1 & 1 \\ \hline 3 & 1 & 1 & 1 \\
\hline 5 & 2 & 2 & 2 \\ \hline 7 & 8 & 7 & 5 \\ \hline 9 & 42 & 32 &
18 \\ \hline 11 & 262 & 175 & 75 \\ \hline 13 & 1828 & 1083 & 383 \\
\hline 15 & 13820 & 7342 & 2850 \\ \hline 17 & 110954 & 53372 & 14984 \\ \hline
\end{tabular} 
\caption{Numbers of Sturm permutations $\pi=\pi_f\in S_n$ for odd $n=1,\cdots,17$.} \label{ta:1} 
\end{table}
%%%%%

\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,14.5)% 
\centerline{\includegraphics[scale=0.9]{Fig/02.eps}}
\end{picture}% 
\caption{All sixteen Sturm attractors ${\cal A}_f$ with $n=9$
equilibria. The corresponding Sturm permutations $\pi_f$ are indicated
in cycle notation.} \label{Fig2.2}
\end{figure}

%%%%%
%
\begin{figure}%[t]
\vspace{6cm}
\begin{picture}(13.5,3.5)% 
\centerline{\includegraphics[scale=1.0]{Fig/03.eps}}
\end{picture}% 
\caption{$C^0$ orbit equivalent Sturm attractors ${\cal A}_f$ with
nonconjugate permutations $\pi_f.$ Note how the attractors differ in
the geometry of the fast unstable manifolds, indicated by double
arrows.} \label{Fig2.3}
\end{figure}


We conclude this section with a short summary of the intriguing
results by H\"arterich on the Sturm attractors $\cal{A}_\varepsilon$
of viscous approximations to nonlinear {\em hyperbolic balance laws;}
see \cite{hae98a}. For scalar equations in one space
dimension such viscous approximations take the form
% 
\begin{eqnarray}u_t+F(u)_x +G (u) =\varepsilon u_{xx}, \label{2.2.25}\end{eqnarray} 
This fits in the framework (\ref{2.2.1}), (\ref{2.2.2}) with
$f:=F'(u)u_x + G(u),$ under Neumann boundary conditions on the unit
interval. The object of interest is the limiting behavior of the Sturm
attractor ${\cal A}_\varepsilon$, for vanishing viscosity $\varepsilon
\searrow 0.$ H\"arterich has derived sufficient conditions on $F,G$
such that
%%%
\begin{itemize} \item[(i)] $\ell_\varepsilon := \dim {\cal A}_\varepsilon$
\item[(ii)] $n_\varepsilon := $ the number of equilibria of ${\cal
A}_\varepsilon$
\item[(iii)] $\pi_\varepsilon$ := the Sturm permutations of ${\cal
A}_\varepsilon$ 
\item[(iv)] ${\cal A}_\varepsilon$ up to orbit equivalence
\end{itemize}

\noindent all stabilize, that is, eventually become independent of
$\varepsilon$ for $0<\varepsilon < \varepsilon_0.$ In view of the
Chafee-Infante example (\ref{2.2.21c}), which is equivalent to
(\ref{2.2.25}) with $F=0,$ cubic $G$, and $\varepsilon :=1/\mu,$ this
is surprising. Indeed $\ell_\varepsilon
=1+[(\pi\sqrt{\varepsilon})^{-1}]$ and
$n_\varepsilon=2\ell_\varepsilon+1$ grow unboundedly for $\varepsilon
\searrow 0,$ in this case.

In \cite{hae98a} sufficient conditions for a uniform dimension bound
$\ell_\varepsilon\leq \ell_0$ were derived, for $\varepsilon\searrow
0.$ Specifically, if $F$ possesses only nondegenerate critical points
and $G$ does not vanish at any critical point of $F,$ then there exist
numbers $\ell_0\in \mathn$ and $\varepsilon_0>0$ such that
$\ell_{\varepsilon}\leq \ell_0$ for all $0<\varepsilon\leq
\varepsilon_0$.


In \cite{hae97} these conditions were refined to imply stabilization of
$\pi_\varepsilon,$ (iii).  Note that (iii) implies stabilization (ii)
of $n_\varepsilon$, trivially, and stabilization (i) of
$\ell_\varepsilon=\dim {\cal A}_\varepsilon,$ by our explicit
representation (\ref{2.2.18}) for the Morse indices $i_k.$
Specifically, there exists an open class of functions $F$ and $G$
(with respect to the strong Whitney topology) such that the following
holds: There exist numbers $\ell_0\in \mathn$ and $\varepsilon_0>0$
such that $\ell_{\varepsilon}\in \{\ell_0-1, \ell_0\}$ for all
$0<\varepsilon\leq \varepsilon_0$.  Explicit conditions on $F$ and $G$
can also be stated which can be verified in concrete situations. Note
that it is not clear whether $n_{\varepsilon}$ stabilizes for
$\varepsilon\to 0$.

If $F\in C^2$ is strictly convex and $G\in C^1$ possesses only simple zeros
that do not coincide with the critical point of $F$, then all four
stabilization properties (i)--(iv) hold, as was claimed above; see
 \cite{hae99b}.

                                                           
The proofs of these results are based on a very careful singular
perturbation analysis of the ODE equilibrium problem, for $\varepsilon
\searrow 0$, including solutions of canard type.

For another example of a singular perturbation of Sturm permutations
and Sturm attractors, involving $x$-dependent nonlinearities
$f=f(x,u)$ of cubic Chafee-Infante type in $u$, see \cite{fienochmal}
and the references there.


%\setcounter{equation}{0}
\subsection{Sturm attractors on the circle}
\label{s-2.3}
In Section \ref{s-2.1}, (\ref{2.1.6})--(\ref{2.1.8}) we have seen how the case
$x\in S^1=\mathr/\MathZ$ of periodic boundary conditions for the
scalar reaction-advection-diffusion equation
% 
\begin{eqnarray}u_t &=& u_{xx}+f(x,u,u_x) \label{2.3.1}\end{eqnarray} 
is a hybrid between one- and two-dimensional domains $\Omega$. The
gradient-flow property fails, due to rotating waves which arise in the
$S^1$-equivariant case $f=f(u,u_x)$. The Sturm property on the other
hand prevails; see Proposition \ref{p-2.1}. At the end of Section
\ref{s-2.1} we have used the Sturm property, only, to prove a
gradient-flow feature of (\ref{2.3.1}) on the interval: convergence to
equilibrium for any bounded solution. We will first indicate the fate
of this convergence property on the circle domain. We will then
present an analogue of Theorems \ref{t-2.3}, \ref{t-2.4} for
heteroclinic orbits between rotating waves.

\subsubsection{Poincar\'e-Bendixson theory}
\label{s-2.3.1}
Intuitively we can say that the dynamics of (\ref{2.3.1}) is
``essentially one-dimension\-al'', asymptotically for any single
trajectory $u(t)={\cal T}(t)u_0$ and under separated boundary
conditions. Indeed, $u(t)$ converges to a single equilibrium as would
be the case for any scalar ODE ${\dot{\bf u}}=g({\bf u}),\; {\bf
u}\in\mathr.$ Still the global attractor ${\cal A}_f$ may be of
arbitrarily large dimension $\ell$, as we recall from the
Chafee-Infante example (\ref{2.2.21c}).

The dynamics of (\ref{2.3.1}) on the circle $x\in S^1$ is then
``essentially two-dimension\-al'', asymptotically for $u(t)={\cal
T}(t)u_0.$ The analogous convergence statement is given by the
Poincar\'e-Bendixson theorem, well known from ODE dynamics ${\dot{\bf
u}}=g(\bf{u})$ in the phase plane ${\bf u}\in\mathr^2$. Of course, the
Chafee-Infante example persists under periodic boundary conditions, if
we reflect solutions through their Neumann boundary. Therefore, the
global attractor may still be of arbitrarily large dimension.

For a precise statement of our Poincar{\'e}-Bendixson result we fix
any $x_0\in S^1,$ arbitrarily, and consider the evaluation projection
% 
\begin{eqnarray}\begin{array}{rl} P:  X &\rightarrow \mathr^2\\
u & \mapsto (u(x_0),u_x(x_0)). \end{array} \label{2.3.2}\end{eqnarray}
The following theorem has been proved in \cite{fiemal89b}; for a
related result obtained independently see \cite{nadira90}. Recall that
${\cal E}=\{u_t=0\}$ denotes the set of equilibria.

%%%%%%%
\begin{Theorem}\label{t-2.6} Fix any $x_0\in
C^1$. Let $u_0\in X\hookrightarrow C^1$ possess a uniformly bounded
solution $u(t)={\cal T}(t)u_0 \in X,t\geq 0,$ of (\ref{2.3.1}) with
$\omega$-limit set $\omega(u_0)$.

Then the evaluation projection $P:\omega(u_0)\rightarrow \mathr^2$ is
injective on the $\omega$-limit set $\omega(u_0)$. Moreover the
following alternative holds:
%%%
\begin{itemize} \item[(i)] either $\omega(u_0)$ is a single periodic orbit, or else
\item[(ii)] both the $\alpha$-limit set and the $\omega$-limit set of
any nonequilibrium solution ${\tilde u}(t),t\in \mathr,$ in
$\omega(u_0)$ consist entirely of equilibria.
\end{itemize}
\end{Theorem}

To indicate the relation of Theorem \ref{t-2.6} with the Sturm
property, Proposition \ref{p-2.1}, we show injectivity of the
restricted evaluation projection
% 
\begin{eqnarray} 
P:\clos ({\rm orb} (\tilde{u}_0))\rightarrow \mathr^2 
\label{2.3.3}\end{eqnarray} 
only along the $X$-closure of any single trajectory ${\rm orb}
(\tilde{u}_0) := \{{\cal T}(t)\tilde{u}_0;\;t\in\mathr\}$ within the
original $\omega$-limit set: $\tilde{u}_0\in \omega(u_0).$ Invoking
the Jordan curve theorem, it is then easy to show that $\clos({\rm
orb}(\tilde{u}_0))$ contains a periodic orbit, if it does not contain
any equilibrium. See for example \cite{fiedle89}. The fine tuning which leads
to Theorem \ref{t-2.6} is much more involved, see \cite{fiemal89b}.

To prove restricted injectivity (\ref{2.3.3}), choose any two distinct
initial conditions $\tilde{u}_0^1, \tilde{u}_0^2$ in $\clos({\rm
orb}(\tilde{u}_0)).$ Suppose that $P\tilde{u}_0^1=P\tilde{u}_0^2,$
proceeding indirectly. Denoting solution curves through
$\tilde{u}^\iota_0,\;\iota=1,2,\;$ by $\tilde{u}^\iota (t),t\in\mathr,$
we then see that
% 
\begin{eqnarray}t \mapsto z(\tilde{u}^1(t) - \tilde{u}^2 (t)) \label{2.3.4}\end{eqnarray} 
drops strictly at $t=0$, by Sturm proposition
\ref{p-2.1}(ii). Choosing $t_0$ and $t_0+\vartheta$ large enough,
(\ref{2.3.4}) implies that
% 
\begin{eqnarray}t \mapsto z(\tilde{u}(t+t_0+\vartheta) -
\tilde{u}(t+t_0)) \label{2.3.5}\end{eqnarray}  
also drops, for some $|t|\leq 1.$ Indeed, the $x$-profile
$\tilde{u}^1(t)- \tilde{u}^2(t)$ possesses only simple zeros in $x$
before and after dropping, and $\tilde{u}^1_0, \tilde{u}^2_0 \in \clos
({\rm orb}(\tilde{u}_0))$ in the topology of $X\subseteq C^1.$ Since
$\tilde{u}_0 \in\omega(u_0),$ the dropping in (\ref{2.3.5}) implies
that there also exists an increasing sequence $t_n\nearrow \infty$ of
dropping times $t=t_n$ of
% 
\begin{eqnarray}t \mapsto z (u(t+t_0+\vartheta) - u(t+t_0)),
\label{2.3.6}\end{eqnarray}   
for the original solution $u$. This contradicts the finiteness
property (iii) of the Sturm Proposition\ref{p-2.1}, and hence proves our
injectivity claim (\ref{2.3.3}) indirectly.

Similar Poincar\'e-Bendixson theorems have been proved to hold for
{\em differential delay equations}
% 
\begin{eqnarray}
\dot{u}(t) &=& f(u(t),u(t-1)) 
\label{2.3.6b}\end{eqnarray} 
under a positive or negative monotonicity assumption on $f$ in the
delayed feedback argument $u(t-1);$ see \cite{mallet88,malsmi90}. For
discrete analogues, the so-called {\em cyclic monotone feedback}
systems, see \cite{smith95} as well as \cite{malsel96a} for
combinations of both structures. The approach in \cite{fiemal89b} is
axiomatic, and includes Jacobi systems
(\ref{2.2.16}).

It is interesting to note that (\ref{2.3.6b}) under positive feedback,
where $f(u,\cdot)$ is strictly increasing, does not exhibit
asymptotically stable periodic solutions. This is reminiscent of the
Hirsch theorem on the absence of stable periodic orbits for monotone
dynamical systems, as discussed in Section \ref{s-2.1}. For negative
feedback, in contrast, stable periodic solutions of (\ref{2.3.6b}) do
arise; see for example \cite{dieketal95} and the references there. As a
curiosity, we note here the twisted periodic boundary condition
% 
\begin{eqnarray}\begin{array}{rcl} u(t,1) & = & -u(t,0) \\ 
u_x(t,1) & = & -u_x(t,0) \end{array}\label{2.3.6c}\end{eqnarray} for
which all the above results remain valid, due to the Sturm
structure. The monotonicity property fails, however: although formally
in effect, it is prevented by the twisted boundary condition
(\ref{2.3.6c}). Indeed $z(\tilde{u}-u) \geq 1$ for any two distinct
solution profiles $\tilde{u},u$ satisfying (\ref{2.3.6c}), as soon as
$t >0.$ This phenomenon is closely related to the Smale objection to
monotonicity, indicated in Section \ref{s-2.1}. Explicit examples of
odd rotating wave solutions $u=u(x-ct)$ for suitable $f$ show that
asymptotically stable periodic orbits in fact do occur under the
twisted boundary condition (\ref{2.3.6c}).  Unfortunately this is a
purely mathematical observation, at the moment: we are not aware of
any applied relevance of our twisted periodic boundary condition.

\subsubsection{Heteroclinic connections of rotating waves}
\label{s-2.3.2}
When we now attempt to describe the Sturm attractors ${\cal A}_f$
under the usual periodic boundary conditions $x\in S^1=\mathr/\MathZ,$
the Poincar\'e-Bendixson Theorem \ref{t-2.6} encourages us to pursue
the program outlined in questions {\bf Q0} -- {\bf Q4} of Section
\ref{s-2.2}. Of course we will have to replace the word ``equilibria''
in {\bf Q0}, {\bf Q1} and address ``equilibria and time periodic
orbits'' as well as their heteroclinic orbits, instead. A fundamental
obstacle, however, arises immediately: the loss of the Morse-Smale
transversality property. In fact, nontransverse intersections of
stable and unstable manifolds may arise. Specifically, any autonomous
planar ODE vector field can be embedded into (\ref{2.3.1}) for a
suitable choice of the nonlinearity $f=f(x,u,u_x),\; x\in S^1$; see
\cite{sanfie92} and \cite{brposa92}. The dynamics is supported, for example,
on a time invariant linear subspace spanned by the spatial Fourier
modes $\sin x$ and $\cos x$. Embedding any planar homoclinic orbit
then immediately shows that the Morse-Smale property may fail, even
when all equilibria and periodic orbits are hyperbolic. As an aside we
mention that time periodic nonlinearities $f=f(t,x,u,u_x),\; x\in
S^1,$ can give rise to transverse homoclinic orbits and hence to Smale
horseshoes with Bernoulli type shift dynamics, by similar
constructions; see \cite{sanfie92}.

A second difficulty of periodic boundary conditions lies in the
combinatorial description of ${\cal A}_f.$ The Fusco-Rocha definition
(\ref{2.2.9}), (\ref{2.2.10}) of the Sturm permutation $\pi_f,$ by
ordering of the boundary values $u,$ clearly depends on the separated
nature of the boundary conditions at $x=0,1.$ The Sturm permutation
probably would have to be replaced by the braid type of the braid in
$(x,u,u_x)\in S^1\times \mathr^2,$ which is defined by the
$x$-profiles of all equilibria and time periodic orbits. Any periodic
orbit $u(t)$ in $X$ can in fact be represented by any snap-shot
$u(t_0)$ without changing the braid type. In the case of Neumann
boundary conditions the braid embeds into a surface: the shooting
surface of solution of (\ref{2.2.24}) with $U_x (0)=0.$ The braid type
is then in fact determined by the Sturm permutation $\pi=\pi_f$.

The above remarks sufficiently demonstrate our lack of understanding
of the general case $f=f(x,u,u_x).$ We therefore now restrict to the
$S^1$-equivariant case of $x$-independent nonlinearities
% 
\begin{eqnarray}u_t=u_{xx} + f(u,u_x),\quad x\in S^1.\label{2.3.7}\end{eqnarray} 
From Section \ref{s-2.1}, (\ref{2.1.7}), (\ref{2.1.8}) we recall that rotating
waves $u(t,x)=U(x-ct)$ arise in that case. More precisely it was
proved in \cite{angfie88} that all periodic orbits $U(t,x)$ are rotating
waves $U=U(x-ct).$ This follows from Sturm Proposition \ref{p-2.1}: the zero
number $z$ must be constant on the span of $U_t$, $U_x$, and hence $U_t$
and $U_x$ must indeed be linearly dependent. The Sturm attractor
${\cal A}_f$ then consists of equilibria ${\cal E}$, rotating waves
${\cal R}$, and their heteroclinic orbits ${\cal H}$:
% 
\begin{eqnarray}
{\cal A}_f &=& {\cal E} \cup {\cal R} \cup {\cal H}.
\label{2.3.8}\end{eqnarray}
We assume ${\cal E}$ and ${\cal R}$ to be {\em normally hyperbolic},
admitting at most a single trivial eigenvalue (or Floquet exponent)
resulting from $x$-shift.

To formulate our result on heteroclinic connections in ${\cal A}_f$,
we slightly adapt Wolfrum's notation of $z$-adjacency, from
(\ref{2.2.22}). First we represent any nonhomogeneous equilibrium in
${\cal E}$ and any rotating wave in ${\cal R}$ by {\em two} elements $U\in X$
satisfying $U_x=0$ at $x=0.$ This can always be achieved because
${\cal E}, {\cal R}$ are invariant under $x$-shift, by
$S^1$-equivariance of (\ref{2.3.7}). The {\em two} representatives are
chosen to attain their maximum and their minimum at $x=0$,
respectively. We then define $z$-order and $z$-adjacency as in
(\ref{2.2.22}), based on the ordering and adjacency properties of
these representatives, at $x=0.$

We also adapt the Morse index $i$ of $U\in {\cal E} \cup {\cal R}$ to
indicate the strong unstable dimension. For equilibria, $i$ counts the
total algebraic multiplicity of complex eigenvalues $\lambda$, after
linearization, with strictly positive real part.  For rotating waves
$U$ we similarly count the nontrivial Floquet multipliers $\lambda$
strictly outside the unit circle.

The following theorem is due to \cite{firowo} and assumes normal
hyperbolicity; see (\ref{2.3.8}).

%%%%%%th2.7
\begin{Theorem}\label{t-2.7} 
Let $U_j,U_k$ be equilibria or rotating waves with strong unstable
dimensions $i_j,i_k$ in the Sturm attractor of the $S^1$-equivariant
reaction-advection-diffusion equation (\ref{2.3.7}) on the circle
$x\in S^1.$ Let $z:=z(U_j-U_k).$ Then there exists a heteroclinic
orbit from $U_j$ to $U_k$ if, and only if, $i_j>i_k$ and $U_j$ and
$U_k$ are $z$-adjacent.
\end{Theorem}

The proof of this theorem at present involves four steps, which we call \\\\
\begin{tabular}{rrrl} &&(i) & transversality\\
&&(ii) & freezing\\
&&(iii) & symmetrization\\
&&(iv) & Neumann embedding
\end{tabular} \\\\


We only briefly mention the topics involved. In step (i), we slightly
extend results by \cite{oliva02} to show that the Morse-Smale property
of transverse intersections of unstable and stable manifolds holds, in
the $S^1$-equivariant case, for normally hyperbolic equilibria and
rotating waves. This allows us, in step (ii), to construct a homotopy
$f^\tau$ of $f=f^0$ which ``freezes'' all rotating waves:
% 
\begin{eqnarray}
f^\tau &:=& f(u,u_x) + \tau c u_x
\label{2.3.8b}\end{eqnarray}  
Here $c=c(u,u_x)$ is chosen to coincide with the constant wave speed
along the $x$-profile $(u,u_x)(x)$ of any rotating wave of $f$,
without introducing any additional rotating waves for any $0\leq \tau \leq
1.$ In effect, this homotopy reduces the speed $c_0$ of any rotating
wave at $\tau=0$ to
%
\begin{eqnarray}
c_\tau &=& (1-\tau)c_0.
\label{2.3.9}\end{eqnarray}
By $\tau=1,$ any rotating wave speed has been tuned down to zero, and
all periodic orbits have become spatially nonhomogeneous equilibria:
${\cal R}=\emptyset$ in (\ref{2.3.8}). Step (iii) performs a second
homotopy which keeps all (frozen) equilibria normally hyperbolic and
reduces $f=f(u,u_x)$ to a nonlinearity which is even in $u_x$ and
hence commutes with the reflection $x\mapsto 1-x.$ In particular, any
equilibrium solution $U$ becomes reflection symmetric with respect to
any of its local maxima or minima, by this $x$-reversibility.
Shifting any equilibrium $U\in {\cal E}$ by some suitable value $x_0$,
we may therefore assume $U_x=0$ at $x=0$ and $x=1/2.$ In step (iv) we
observe that all heteroclinic orbits of Theorem \ref{t-2.4} are then
in fact already represented by reflection symmetric heteroclinic
orbits. The latter are already known: we only have to apply Wolfrum's
Theorem \ref{t-2.4}. to the resulting Neumann problem on the
half-interval $0\leq x \leq 1/2.$ This completes our sketch of a proof
of Theorem \ref{t-2.7}. For complete details see \cite{firowo}.  


We conclude this section with a few remarks concerning related
results. The results by Hale and Raugel concerning upper and lower
semicontinuity of attractors ${\cal A}_f$ for {\em thin domains}
\cite{hara92} allow us to fatten the circle $\Omega_0=S^1$ to thin
annuli $\Omega_\varepsilon =S^1 \times(-\varepsilon,\varepsilon )$ and
recover all dynamics of ${\cal A}_f$ on $\Omega_\varepsilon.$ More
generally domains $\Omega_\varepsilon =S^1 \times
\tilde{\Omega}_\varepsilon$ work, for suitably ``small'' cross
sections $\tilde{\Omega}_\varepsilon$ in terms of a sufficiently large
second eigenvalue of the Neumann problem on the cross section
$\tilde{\Omega}_\varepsilon$. Similarly, nonlinearities $f+\varepsilon
\tilde{f}$ can be allowed to depend on $x,$ slightly, via
$\tilde{f}(x,u,u_x)$ and small $\varepsilon$. In the spirit of
\cite{fievis01b,fievis01a}, nonlinearities
$f=f(\varepsilon,x/\varepsilon,u,u_x)$ are also admissible, under
Diophantine quasiperiodicity conditions on the rapid spatial
dependence $x/\varepsilon$ of $f.$

Fast travelling waves in cylinder domains $x=(\xi,\eta)\in
\Omega=S^1\times \mathr$ provide another interpretation of our
heteroclinic orbits. Consider travelling wave solutions
% 
\begin{eqnarray}u(t,x) &=& U(t- \varepsilon\eta, x)\label{2.3.10}\end{eqnarray} 
of a scalar $x$-independent parabolic equation (\ref{1.1.1}). For
large wave speeds $1/\varepsilon$ in the unbounded $\eta$-direction, we
obtain
% 
\begin{eqnarray}U_\tau &=& U_{\xi\xi}+\varepsilon^2 U_{\tau\tau}+f(U,U_\xi,-\varepsilon U_\tau),
\label{2.3.11}\end{eqnarray}  
abbreviating $\tau :=t-\varepsilon\eta.$ In \cite{ScheelDiss} it was shown
how to omit both $\varepsilon$-terms, in the fast wave speed limit
$\varepsilon\searrow 0.$ See also \cite{CaMoSo} for nonlinearities $f$
which do not depend on the gradient term $U_\xi.$ Invoking Theorem
\ref{t-2.7} then provides heteroclinic orbits $U(\tau,\xi)$ between
rotating waves $U_1(\xi-c_1\tau), U_2(\xi-c_2\tau),$ typically
rotating at different angular speeds $c_1,c_2.$ Clearly $U(\tau,\xi)$
then describes a nonmonotone wave $u$ propagating rapidly along the
cylinder axis $\eta$ of $\Omega,$ and connecting $\eta$-asymptotic
states which rotate at different angular $\xi$-speeds $c_1,c_2.$ The
complexity of such wave profiles is in marked contrast with the
monotone travelling waves found by Berestycki and Nirenberg using
comparison methods \cite{BeNi}. See also the beautiful complementary
approach by a variational characterization in \cite{Hei}.

Returning to one-dimensional $x$ we conclude this section by
indicating why results like Theorems \ref{t-2.2.1} -- \ref{t-2.7} do
not carry over to systems $u\in\mathr^N,N=2,$ not even in a single
space dimension. Following \cite{fiepol90}, we consider
% 
\begin{eqnarray}u_t &=& u_{xx} + f(x,u)+c(x)\int^1_0 u\, \rmd x, 
\label{2.3.12}
\end{eqnarray} 
only for scalar $u\in\mathr,$ but involving a linear nonlocal term
$c(x)\int udx.$ We consider $0<x<1$ with Dirichlet boundary
conditions. It has then been proved that very general
finite-dimensional vector fields $\dot{\bf{u}}=g(\bf{u}),\;
\bf{u}\in\mathr^n$ arise, on an invariant (center) manifold of
(\ref{2.3.11}) and up to any finite polynomial order, for suitable
choices of $f,c.$ The constraints on $g(\bf{u})$ are rather mild:
$g(0)=0$ and the eigenvalues of $g'(0)$ on the imaginary axis should
be simple. In particular this includes the possibility of complicated
dynamics of Bernoulli shift type. These results are very similar, in
spirit, to earlier analogous observations by Hale for retarded
functional differential equations; see \cite{ha85}.

Similar remarks apply to the case of scalar equations (\ref{1.1.1}),
$u\in\mathr^N,N=1,$ in several space dimensions. Dancer and
Pol\'a\v{c}ik \cite{dapo02} have proved that a $C^1$-dense set of finite-dimensional
vector fields $\dot{\bf{u}}=g(\bf{u})$, $\bf{u}\in\R^{\bf{n}}$ can be
realized on an invariant manifold, for suitable choices of
$x$-independent reaction-advection nonlinearities $f=f(u,\nabla u)$
and of two-dimensional smooth bounded domains $\Omega \subset
\mathr^2$. For suitable $f=f(x,u,\nabla u)$ with
only linear dependence on $\nabla u$ and arbitrary, but fixed domains
$\Omega \subset \mathr^2,$ a similar result is due to \cite{polaci95},
\cite{priryb98a,priryb98b}. The gradient-like case $f=f(x,u)$ on a
two-dimensional ball allows embeddings of $C^1$-dense sets of gradient
vector fields $g({\bf u})=-\nabla_{{\bf u}} G({\bf u}).$ See also the
survey \cite{polaci02}.

These results certainly motivate a quest for more refined structural
conditions on nonlinearities $f$, as well as domains $\Omega,$ which
impose incisive restrictions on their resulting spatio-temporal
dynamics.


\section{One unbounded space-dimension: travelling waves}
\label{s:1u}
In this chapter, we address some aspects of the dynamics of
reaction-diffusion systems in large or unbounded, one-dimensional
domains. Motivated by the motion of layers and fronts, we study
essential spectra that arise in the linearization about travelling
wave solutions, Section~\ref{s:1}. In Section~\ref{s:1.3}, we extend
this linear analysis to a nonlinear bifurcation result. Depending on
Fredholm indices in the essential spectrum, we find existence or
nonexistence of periodic orbits in a Hopf bifurcation caused by the
essential spectrum. We conclude this chapter with an investigation of
the limiting behavior of the spectrum of the linearization about a
travelling wave, when the domain size tends to infinity. Under
separated boundary conditions, the limiting spectrum differs from the
spectrum in the unbounded domain. The continuous parts of the limiting
spectrum consist of a finite collection of curves which we call the
absolute spectrum.

\subsection{Unbounded domains and essential spectra}
\label{s:1}
\subsubsection{From bounded to unbounded domains}
\label{s:1.1}
The results on global attractors in scalar reaction-diffusion
equations reviewed in the preceding sections show that certain
dynamical properties do not depend on boundary conditions. The Sturm
nodal property prevails, for example, independently of the boundary
conditions. This allows for homotopies between Neumann, Dirichlet,
mixed, and sometimes even periodic boundary conditions.  An extreme
statement in this direction is that the class --- in contrast to
individual global attractors --- of all Sturm attractors ${\texttt
A}^P$ does not depend on the type of (separated) boundary conditions;
see Section \ref{s-2.2}.

In descriptions of phenomena in experiments, it is often quite
desirable to separate the influence of the boundary from what we call
the {\em inner dynamics} of a reaction-diffusion system. See for
example~\cite{deKepper} for a critical view on the influence of
boundary conditions in chemical experiments.  Mathematical statements
on the influence of boundaries can be found mainly in the context of
singular perturbation theory. For an illustration, we rewrite the
Chafee-Infante problem~(\ref{2.2.21c}) as
\[
u_t = \eps^2 u_{xx} + u ( 1-u) (1+u),\qquad x\in (0,1),
\]
say with Neumann boundary conditions. Rescaling $x$, the parameter
$\eps$ can be interpreted as a measure for the size of the domain,
$0\leq x \leq L=1/\eps$. For small $\eps$, the motion on the unstable
manifolds of equilibria (which constitute the Chafee-Infante
attractor, by (\ref{2.2.7})) becomes exponentially slow in the
parameter $\eps$; see~\cite{Fus,Car}. ``Most'' of the time, solutions
consist of a finite collection of transition layers at $x=x_j(t)$ of
the approximate local form
\begin{equation}\label{e:layshap}
u_j\sim\pm\tanh(\frac{x-x_j(t)}{\sqrt{2}\eps}).
\end{equation}
The exponentially slow motion $x_j(t)$ is driven by the inner
dynamics, caused by interaction and annihilation of layers, and the
influence of the boundary. The strength of the interaction and the
influence of the boundary indeed decrease exponentially with distance:
\[
|\dot x_j(t)|\leq C\exp(-C' (\inf_{j'\not=j}\{|x_j-x_{j'}|/\eps\}+
\min\{|x_j|,|1-x_j|\})).
\] 
For most initial conditions, the slow motion eventually leads to
annihilation of all layers and we recover the Hirsch result
\cite{hirsch88} on generic convergence to stable equilibria in
monotone dynamical systems, $u_\pm(x)\equiv \pm 1$ in our case ---
however only after time spans which may well exceed realistic
experimental conditions.

The limiting case $\eps=0$ is best described in the scaling
\[
u_t=u_{xx} + u(1-u)(1+u), \qquad x\in (-L,L),
\]
with $L=1/\eps$. In the limit $L=\infty$, we are led to consider the
unbounded real line as an idealization describing the inner dynamics.

One important but largely unresolved question concerning the limit $L\rightarrow \infty$ of increasing domain size, is in how far energy considerations can be localized, separating effects of the boundary from the intrinsic dynamics. Energy, alias the Lyapunov function in (\ref{2.1.3}), is a nonlocal function and need not even be finite in an unbounded domain. Gallay and Slijep\v{c}evi\v{c} \cite{gasl02} show how this fact may lead to recurrent, quite unexpected behavior in ``gradient'' systems.

Unbounded domains feature yet another phenomenon: {\em non-compact
symmetry}. Reaction-diffusion systems (\ref{1.1.1}) with
$x$-independent reaction term $f=f(u,\nabla u)$, are invariant under
spatial translation $x\to x+g$: if $u(t,x)$ is a solution, so is
$u(t,x+g)$ for any $g\in\R$, fixed. The slow motion $x_j(t)$ of layers
in the Chafee-Infante problem reflects this translational
symmetry. The time evolution is ``composed'' of translates of the single
layer (\ref{e:layshap}). Whereas the role of the noncompact symmetry
group of translations $g\in\R$ is obvious, here, Euclidean symmetry $G=SE(2)$ will
play a major and more subtle role in the discussion of spiral
wave dynamics in two-dimensional domains; see Chapter~\ref{s:4}.

Separated boundary conditions break this translational
equivariance. Periodic boundary conditions, in contrast, preserve the
action of at least a continuous, compact group $SO(2)=\R/\Z$ as a
remaining symmetry, when the real line is truncated to a bounded
interval; see Section~\ref{s-2.3}. Still, even periodic boundary
conditions fail to mimic the translational drift along non-compact
group orbits like the observed motion of a single stable front: under
periodic boundary conditions, fronts come in pairs and typically
annihilate or strongly interact after some finite time interval. We
will further investigate the role of boundary conditions in the
presence of transport and drift in Section \ref{s:2}.

If the domain is the real line $x\in\R$, the discussion of equilibria
in $u_t=u_{xx}+ f(u,u_x)$ has at least to be augmented to include
travelling waves, $u(t,x)=q(x-ct)$, $c\in\R$, with $\sup_\xi
u(\xi)<\infty$, just like in the case of periodic boundary conditions,
Section~\ref{s-2.3}. Travelling waves are a special case of {\em
relative equilibria}, where time evolution of a profile is described
by motion along the group action, here, translated states. We discuss
relative equilibria more generally in the context of spiral wave
dynamics in Chapter \ref{s:4}. {\em Fronts} are special travelling
waves, where $q(\xi)$ possesses a heteroclinic asymptotic behavior for
$\xi\to\pm\infty$, for example $q(\xi)\to q_\pm$ for
$\xi\to\pm\infty$. A particular case are the layers in the
Chafee-Infante problem, where $c=0$.

\subsubsection{Spectra of  travelling waves: group velocities and
Fredholm indices}
\label{s:1.2}
For a scalar equation $u_t=u_{xx}+ f(u)$, travelling waves with
$c\not= 0$ are either spatially constant, $u(\xi)\equiv u_0$, or
heteroclinic front solution. This is due to the gradient-like
structure of the travelling-wave ordinary differential equation
\[
u_\xi = v,\quad v_\xi = -cv - f(u).
\]
More generally, we address {\em reaction-diffusion systems}
\begin{equation}\label{e:rdsys1}
u_t = Du_{xx} + f(u), \qquad 
\end{equation}
with $x\in\R$, $u\in\R^N$ and a positive, diagonal diffusion matrix
$D={\rm diag\,}(d_j)>0$. Systems of several reaction species, $N>1$,
in general, neither possess gradient-like structure, nor monotonicity
properties --- or even nodal properties --- in the sense of
Section~\ref{s-2.1}. Still, heteroclinic travelling-wave solutions
$u(t,x)=q_*(x-ct)$ remain a fundamental, and elementary, ingredient to
their dynamics. 

Given such a travelling wave, we ask for its
stability. We therefore consider the reaction-diffusion system in a
comoving frame $\xi=x-ct$,
\begin{equation}\label{e:rdsys1c}
u_t = Du_{\xi\xi}  +c u_\xi+ f(u), \qquad x\in\R,
\end{equation}
where the travelling wave becomes an equilibrium $q_*(\xi)$.
Throughout, $x$ always refers to the spatial coordinate in the steady
frame, whereas $\xi$ always refers to a comoving frame with speed
$c\geq 0$. The case $c<0$ is obtained reflecting $x\to -x$. The
travelling wave moves towards $x=+\infty$ and, in the comoving frame,
the reaction-diffusion system inherits a drift term $cu_\xi$,
representing transport towards $x=-\infty$.  In the comoving frame, we
linearize at $q_*$ to find
\begin{equation}\label{e:rdsys1l}
w_t = D w_{\xi\xi} + c w_\xi+ f'(q_*(\xi)) w =: {\cal L}_* w.
\end{equation} 
For simplicity, we first consider the operator ${\cal L}_*$ as a
closed, unbounded operator on $L^2(\R)$. All results of this section
remain valid if we replace the function space $L^2$ by $L^p$, or by
$1<p<\infty$, or bounded, uniformly continuous functions. The {\em
spectrum} ${\rm spec}\,{\cal L}_*\subset\C$ is defined as the set of
$\lambda$ such that ${\cal L}_*-\lambda$ is not boundedly invertible.
Various notions of {\em stability} for individual fronts $q_*$ have
been suggested. We call a travelling wave {\em spectrally stable}, if
the spectrum of the linearized operator ${\cal L}_*$ is contained in
$\{\Re\lambda\leq 0\}$. We caution the reader that spectral stability
need not imply asymptotic stability on our unbounded domain $x\in\R$.

For a refined discussion of stability properties, we decompose
the spectrum into the {\em essential spectrum}
\[
{\rm spec}_{\rm ess}\,{\cal L}_* :=
\{
\lambda\in\C;\; {\cal L}_* -\lambda
\mbox{ is not Fredholm of index 0} \}
\]
and its complementary part, the {\em point spectrum}
\[
{\rm spec}_{\rm pt}\,{\cal L}_* := {\rm spec}\,{\cal L}_*
\setminus{\rm spec}_{\rm ess}\,{\cal L}_*.
\]
The appearance of essential spectrum is caused by the non-compactness
of the real line $x\in\R$, here. By robustness of Fredholm properties,
the essential spectrum is closed. As usual, for $\lambda\in {\rm
spec}_{\rm pt}\,{\cal L}_*$, nontrivial $u\in \Ns({\cal L}_*-\lambda)$
are then called eigenfunctions and algebraic multiplicity is
associated to Jordan blocks. Due to analyticity of the eigenvalue
problem in $\lambda$, the only possible degeneracy in this description
of point spectrum is characterized by connected components of the set
of $\lambda$ such that the Fredholm index of ${\cal L}_*-\lambda$ is
zero and the kernel is nontrivial; see~\cite{Kato, Henry}.

For example, consider the heteroclinic case of a travelling front
$q_*$, where $q_*(\xi)\to q_\pm$ for $\xi\to\pm\infty$. In
Proposition~\ref{p:ivc}, below, we will characterize the essential
spectrum of ${\cal L}_*$ on $L^2(\R)$ in terms of the spectra of the
asymptotic linearizations ${\cal L}_\pm$ at $q_\pm$:
\[
{\cal L}_\pm w := Dw_{\xi\xi} + cw_\xi + f'(q_\pm)w.
\] 
Unlike ${\cal L}_*$, the differential operators ${\cal L}_\pm$ possess
constant, $x$-independent coefficients. The spectra of ${\cal L}_\pm$
are therefore readily computed in terms of exponentials. Setting
$w(\xi)=\rme^{-\nu \xi}w_0$, with $\nu\in\C$, the spectral problem
${\cal L}_\pm w = \lambda w$ is transformed into the {\em complex
dispersion relation} 
\begin{equation}
\label{e:dr} 
0=d_\pm(\lambda,\nu):={\rm det\,}(D\nu^2-c\nu + f'(q_\pm)-\lambda),
\end{equation}
which is polynomial in the complex variables $\lambda,\nu\in \C$. For
fixed {\em spatial eigenvalue} $\nu$, we find $N$ roots
$\lambda_\ell=\lambda_\ell(\nu)$. Conversely, fixing the {\em temporal
eigenvalue} $\lambda$, we find $2N$ roots $\nu_j=\nu_j(\lambda)$. By
Rouch\'e's theorem, the roots depend continuously on $\nu$ or
$\lambda$, respectively. We recover the Fourier transform in $x$ when
specializing to $\nu=\rmi k$ with real spatial wavenumber $k\in\R$. We
therefore conclude that
\[
\lambda\in{\rm spec\,}{\cal L}_\pm\quad \Longleftrightarrow\quad
d_\pm(\lambda,\rmi k)=0 \mbox{ for some } k\in\R.\
\] 
The spectra of ${\cal L}_\pm$ then each decompose into $N$
algebraic curves
\begin{equation}\label{e:fb}
{\rm spec\,}{\cal L}_\pm = \bigcup_\ell \Gamma_\pm^\ell,\quad \Gamma_\pm^\ell =
\{\lambda_\pm^\ell(\rmi k);\;k\in\R\}.
\end{equation}
The imaginary part of the tangent vector to any dispersion curve
$\lambda_\pm^\ell(\rmi k)$, $k\in\R$, is called its {\em group
velocity} $c_{\rm g}$:
\begin{equation}
\label{e:cg}
c_{{\rm g},\pm}^\ell(k)=\Im \frac{\rmd \lambda^\ell_\pm(\rmi k)}{\rmd k}
.\end{equation}
From the dispersion relation (\ref{e:dr}), the sum $c_{{\rm
g},\pm}^\ell(k)+c$ is independent of the speed $c$ of the coordinate
frame. 

For an interpretation, first consider the special case of purely
imaginary temporal eigenvalues $\lambda_\pm^\ell(\rmi k)$. For
classical linear conservative wave equations like $u_t=u_x$,
$u_{tt}=u_{xx}$, the Schr\"odinger equation, etc., the dispersion
curve $\Gamma$ is indeed vertical. Then the group velocity $c_{\rm g}$
describes the speed and direction of $x$-propagation of an initial
condition, which consists of a narrow Gaussian wave package of spatial
oscillations $\rme^{\rmi k x}$. The same observation holds true in the
dissipative context, for $\lambda_\pm^\ell(\rmi k)$ with nonvanishing
real part. The real part $\Re \lambda_\pm^\ell(\rmi k)$ then just
produces a superimposed exponential temporal growth or decay of the
wave packet, which moves along the $x$-axis with its group velocity
$c_{\rm g}$. 

We orient the curves $\Gamma_\pm^\ell$ in the direction of increasing
$k$. Upwards orientation of $\Gamma^\ell_\pm$, in the complex plane,
therefore indicates a positive group velocity, and downwards
orientation a negative group velocity. The following proposition
identifies the curves $\Gamma_\pm^\ell$ as {\em Fredholm borders},
where the Fredholm index of ${\cal L}_*$ changes.
\begin{Proposition}
\label{p:ivc}
The operator ${\cal L}_*-\lambda$ on $L^2(\R)$ is Fredholm for
$\lambda$ in the complement of ${\rm spec\,}{\cal L}_+\cup\, {\rm
spec\,}{\cal L}_-$. The Fredholm
index increases by one upon crossing a curve  $\lambda_+^\ell(\rmi k)$ from left to right, with
respect to its given orientation; it decreases by one
upon crossing a curve $\lambda_-^\ell(\rmi k)$ from left to right.
\end{Proposition}
For example, assume that the group velocity $c_{{\rm g},+}^\ell(k)$ or
$c_{{\rm g},-}^\ell(k)$ is directed towards the front position, which
is stationary in our comoving coordinates: $\pm c_{{\rm
g},\pm}^\ell(k)<0$. Then the Fredholm index of ${\cal L}_*-\lambda$
{\em decreases} when $\Re\lambda$ increases across the spectral curve
$\lambda_+^\ell(\rmi k)$ or $\lambda_-^\ell(\rmi k)$,
respectively. Conversely, if the group velocity is directed away from
the interface, $\pm c_{{\rm g},\pm}^\ell(k)>0$, then the Fredholm
index of ${\cal L}_*-\lambda$ {\em increases} when $\Re\lambda$
increases across the spectral curve $\lambda_+^\ell(\rmi k)$ or
$\lambda_-^\ell(\rmi k)$, respectively; see Figure~\ref{f:cvi}. In
particular, the Fredholm index is $+1$ to the left of the most
unstable curve $\Gamma_\pm^\ell$, with $\Re\lambda$ maximal for all
choices of $\ell,\pm$, if the associated group velocity is directed
towards the front. The Fredholm index  is $-1$ if it is directed away from the front.
\begin{figure}%[!ht]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,4.5)%
\centerline{\includegraphics[scale=0.5]{Fig/index.eps}}
\end{picture}%
\caption[]{A schematic plot of the oriented spectral curves
$\Gamma_\pm^\ell$ and the Fredholm indices of ${\cal L}_*-\lambda$.}
\label{f:cvi} 
\end{figure} 

The proof of Propostion~\ref{p:ivc} is based on a formulation of the
spectral problem $({\cal L}_*-\lambda)u$ as a first-order differential
equation
\begin{equation}\label{e:fsto}
u_\xi=v,\qquad v_\xi=D^{-1}(-cv-f'(q_*(\xi))u+\lambda u).
\end{equation}
Consider therefore a generalized eigenvalue problem for first-order
differential operators
\begin{equation}\label{e:gevp}
{\cal M}(\lambda)\mbf{w}:=\frac{\rmd \mbf{w}}{\rmd\xi}
-\mbf{A}(\xi;\lambda)\mbf{w}=0.
\end{equation}
Assume that the matrices $\mbf{A}(x;\lambda)$ are asymptotically constant:
$\mbf{A}(x;\lambda)\to \mbf{A}_\pm(\lambda)$ for $x\to\pm\infty$. Note
that the asymptotic eigenvalues $\nu^\ell_\pm(\lambda)$ of ${\cal
A}_\pm(\lambda)$ satisfy the dispersion relation
\[
d_\pm(\lambda,\nu^\ell_\pm(\lambda)):={\rm det}\,(-\nu_\pm^\ell -\mbf{A}_\pm(\lambda)) =0.
\] 
We define the asymptotic, {\em spatial Morse indices} $i_\pm(\lambda)$
as the dimensions of the unstable eigenspaces of ${\mbf
A}_\pm(\lambda)$, respectively. Note that from the Ansatz
$\mbf{w}(\xi)=\rme^{-\nu \xi}\mbf{w}_0$, the spatial Morse index is
given by the number of roots $\nu$ of the dispersion relation for
given $\lambda$, with $\Re\nu<0$.
\begin{Proposition}\label{p:fredi}
The operator ${\cal M}(\lambda)=\frac{\rmd}{\rmd \xi}-{\mbf
A}(\xi;\lambda)$, considered as a closed, unbounded operator on $L^2(\R)$, is
Fredholm if, and only if, $\Re\nu^j_\pm\not=0$ for all $j$. The
Fredholm index is given by
\begin{equation}\label{e:fredi}
i({\cal M}(\lambda))=i_-(\lambda) - i_+(\lambda)
,\end{equation}
where the spatial Morse
indices $i_\pm(\lambda) = \# \{\Re\nu^j_\pm <0\}$ count the number of unstable
eigenvalues of the asymptotic matrices $\mbf{A}_\pm(\lambda)$, with
algebraic multiplicity. 
\end{Proposition}
In the formulation as a first order, generalized eigenvalue problem,
the proposition actually covers a much broader class of linear
operators than linearizations about travelling waves in
reaction-diffusion equations. We refer to~\cite{SSmathnach} for a
still more general formulation, which allows to cover linearizations
about time-periodic states as well as problems in several space
dimensions $m>1$.

Essential spectra do not depend on function spaces
such as $L^p$, $C^0$, or $H^k$, as long as translation invariant norms
are considered. However, the locations of the Fredholm borders
$\Gamma^\ell_\pm$ do change when we consider spaces with exponential
weights. For weights ${{\underline{\eta}}}=(\eta_+,\eta_-)\in\R^2$,
define
\begin{equation}\label{e:expweight}
L^2_{\underline{\eta}}:=\left\{u\in L^2_{\rm
loc};\;\|u\|_{L^2_{{\eta}}}<\infty\right\}, \quad
\|u\|_{L^2_{\underline{\eta}}}^2:=\int_{\R^+}|\rme^{\eta_+\xi}u(\xi)|^2
+\int_{\R^-}|\rme^{\eta_-\xi}u(\xi)|^2.
\end{equation} 
Positive weights $\eta_\pm>0$ stabilize transport to the left.  For
example, consider pure transport to the left by $u_t=u_\xi$, on the
entire real line. The trivial solution $q_*\equiv 0$ is then stabilized by an exponential weight with
$\eta:=\eta_+=\eta_->0$. Indeed, the exponentially weighted solution
$v(t,\xi)=\rme^{\eta \xi}u(t,\xi)$ satisfies
$\|u\|_{L^2_{\underline{\eta}}}=\|v\|_{L^2}$, for all fixed $t$. Moreover,
$v$ solves the damped transport
equation $v_t=v_\xi - \eta v$, and hence decays to zero exponentially,
in $L^2(\R)$, with rate $\eta>0$.

More generally, transport is measured by the group velocity,
as reflected by  the following lemma.
\begin{Lemma}
\label{l:cgweight}
The essential spectrum in $L^2_{\underline{\eta}}$ can be determined from
Proposition~\ref{p:ivc}, if we replace the dispersion
relations~(\ref{e:dr}) at the asymptotic states by the shifted
dispersion relations
\begin{equation}\label{e:dre}
d_\pm^{{\eta}}(\lambda,\nu):=d_\pm(\lambda,\nu-\eta_\pm)
.\end{equation}
In particular, small weights $\eta_\pm$ infinitesimally shift the
Fredholm borders $\Gamma^\ell_\pm=\{\lambda_\pm^\ell(\rmi
k;\eta_\pm);\,k\in\R\}$ as follows:
\[
\left.\frac{\partial\Re\lambda^\ell_\pm(\rmi
k;\eta)}{\partial{\eta}}\right|_{\eta_\pm=0}=-c^\ell_{{\rm
g},\pm}(k).
\]
\end{Lemma}
The key observation in the proof of Lemma~\ref{l:cgweight} is, that
$\lambda^\ell_\pm(\rmi k;\eta)=\lambda^\ell_\pm(\rmi k - \eta;0)$, not only
for real but also for complex $k$. Therefore 
\[
\left.\frac{\partial\Re\lambda^\ell_\pm(\rmi
k;\eta)}{\partial{\eta}}\right|_{\eta=0}=
\left.\frac{\partial\Re\lambda^\ell_\pm(\rmi k-\eta;0)}{\partial{\Im
k}}\right|_{\eta=0}=-\left.\frac{\partial\Im\lambda^\ell_\pm(\rmi
k-\eta;0)} {\partial{\Re k}}\right|_{\eta=0}=-c^\ell_{{\rm g},\pm}(k).
\]

Here, we have used the Cauchy-Riemann equations for complex analytic
functions in the second identity.

Proofs of Propositions~\ref{p:ivc} and~\ref{p:fredi} can be found, in
a slightly different context, in~\cite{Palmer,Henry}; see
also~\cite{SSmathnach} for a more general setup. We conclude this
section on essential spectra with several examples.

Consider the linear scalar reaction-advection-diffusion equation 
\begin{equation}\label{e:cd}
w_t=w_{\xi\xi} +  w_\xi + a w =:{\cal L}_* w
\end{equation}
with positive drift velocity and constant reaction coefficient
$a\in\R$. We first view ${\cal L}_*$ as an unbounded operator on
$L^2(\R)$, as in Propositions~\ref{p:ivc} and~\ref{p:fredi}.

From (\ref{e:dr}) and (\ref{e:fb}), we obtain the dispersion relation
and spectrum of ${\cal L}_*$ to be given by the parabola
\begin{eqnarray}\label{e:drcd}
0&=&d(\lambda,\rmi k)=-k^2 - \rmi k + a -\lambda;\\
{\rm spec}\,{\cal L}_* &=& \{\lambda=-k^2 + a + \rmi
k;\,k\in\R\}=
\{\lambda\in\C;\
\,\Re\lambda = (\Im \lambda)^2 + a\} =:\Gamma.\nonumber
\end{eqnarray}
Preparing for the more general case, where $a$ depends on $\xi$, we now
consider (\ref{e:cd}) on $\xi\in\R_+$ or $\R_-$ and equip the arising
differential operators ${\cal L}_>$ and ${\cal L}_<$  with Dirichlet
boundary conditions in $\xi=0$. We claim that
\begin{eqnarray}\label{e:drcdpm}
{\rm spec}\,{\cal L}_> = {\rm spec}\,{\cal L}_< = 
\{\lambda\in\C;\
\,\Re\lambda \leq (\Im \lambda)^2 + a\},
\end{eqnarray}
with Fredholm index $i({\cal L}_>-\lambda)=1$ and $i({\cal
L}_<-\lambda)=-1$ in the interior of the spectrum. Indeed, we follow
(\ref{e:fsto}) to rewrite $({\cal L}_>-\lambda)w=0$ as a first order
system
\begin{equation}\label{fsor}
{\cal M}(\lambda)\mbf{w} = \mbf{w}_\xi + \mbf{A}(\lambda)\mbf{w} = 0,
\qquad \mbf{A}(\lambda)=\left(\begin{array}{cc} 0&-1\\ a -\lambda &
1\end{array}\right),
\end{equation}
for $\xi>0$. For $\Re\lambda<(\Im\lambda)^2+a$, to the left of the
Fredholm border $\Gamma$, both eigenvalues $\nu^{1,2}(\lambda)$ of
$\bf{A}(\lambda)$ possess strictly positive real part.  The
one-dimensional subspace in the $\mbf{w}$-plane, selected by the
Dirichlet boundary conditions at $\xi=0$ therefore gives rise to a
one-dimensional kernel of exponentially decaying functions. The
operator ${\cal L}_>$ is surjective since solutions to the initial
value problem
\[
\mbf{w}_\xi = -\mbf{A}(\lambda)\mbf{w} + \mbf{h}(\xi)
\]
with $\mbf{w}(0)=0$ for $\xi>0$ belong to $L^2$. This shows that
${\cal L}_>-\lambda$ is Fredholm of index 1 to the left of the
Fredholm border. The operator ${\cal L}_<$ is conjugate to the adjoint of
${\cal L}_>$ on $\R^+$, by the reflection $x\mapsto -x$, and therefore
possesses Fredholm index -1. For $\lambda$ to the right of the
Fredholm border, both operators  are invertible
since the Dirichlet subspace does not coincide with an eigenspace of
$\mbf{A}(\lambda)$. 

Next consider piecewise constant coefficients $a(\xi)=a_\pm$ for
$\pm\xi>0$ in the advection-diffusion equation (\ref{e:cd}). We then
obtain corresponding spectral borders $\Gamma_\pm\subset\C$, which are
parabolas shifted by $a_\pm$, respectively. Assume $a_+>a_-$, first. We
claim that 
\begin{eqnarray}\label{e:drcdpm1}
{\rm spec}\,{\cal L}_* =
\{\lambda\in\C;\
\,(\Im \lambda)^2 + a_- \leq \Re\lambda \leq (\Im \lambda)^2 + a_+\},
\end{eqnarray}
with Fredholm index $i({\cal L}_*-\lambda)=1$ in the interior of the
spectrum. Indeed, consider the first-order eigenvalue problem
(\ref{fsor}) with $\lambda$ between the spectral borders
$\Gamma_\pm$. The eigenvalues of the asymptotic matrices $\mbf{A}_\pm$
then satisfy $\Re\nu^1_-<0<\Re\nu^2_-$ and $\Re\nu^{1,2}_+>0$. The
differential equation therefore possesses a one-dimensional subspace of
solutions $\mbf{w}(\xi)$ which are continuous in $\xi=0$ and decay
for  $\xi\to\pm\infty$. This contributes a one-dimensional kernel in the
Fredholm index one region of ${\cal L}_*$. Surjectivity between the
Fredholm borders, and invertibility in the Fredholm index zero region,
follow just like for the operator ${\cal L}_>$. 

In the opposite case $a_->a_+$, the Fredholm index is -1 between the
Fredholm borders $\Gamma_\pm$.  In the interpretation of
Proposition~\ref{p:ivc}, the group velocity $c_{\rm g}=-1$ is
constant and negative on $\Gamma_+$, directed towards the ``front'' in
$\xi=0$. The Fredholm index therefore decreases from left to
right. However,  on $\Gamma_-$ the group velocity $c_{\rm g}=-1$ is
directed away from the ``front'', and the Fredholm index increases
from left to right.

Essential spectra for more general coefficients $a=a(\xi)$ with
$a(\xi)\to a_\pm$  coincide with the essential spectra computed above,
since localized $\xi$-dependence of $a(\xi)$ amounts to a relatively
compact perturbation of ${\cal L}_*$, which does not change Fredholm
properties~\cite{Kato}. 

Just like in the example of simple translation $u_t=u_\xi$,
exponential weights $\underline{\eta}=(\eta_+,\eta_-)$ with
$\eta_+,\eta_->0$ shift the Fredholm borders to the left, as can be
readily seen from the complex dispersion relations, or
Lemma~\ref{l:cgweight}. 

As a final example, we comment on spectra in the case of spatially
periodic coefficients $a=a(\xi)=a(\xi+p)$. This situation arises under
linearization $a(\xi)=f'(q_{\rm wt}(\xi))$ along a periodic wavetrain
with nonlinear spatial wave vector $k_{\rm wt}=2\pi/p$, in comoving
coordinates $\xi$; see for example the end of Section~\ref{s:sdei}. In
a more demanding context, the problem reappears in our stability
analysis of Archimedean spiral waves, for radial spatial dynamics in
the farfield limit of infinite radius; see
Section~\ref{s:2.1a},~\ref{s:3.3.2}, and equations
(\ref{e:5.27})-(\ref{e:cglin0}). Standard spatial Floquet theory for
the eigenvalue problem
\begin{equation}\label{e-fl}
u_\xi=v,\quad v_\xi=-cv-a(\xi)u-\lambda u
\end{equation}
will consider exponential functions $\mbf{w}=(u,v)$,
\begin{equation}\label{e-flth}
\mbf{w}(\xi)=\rme^{-\nu\xi}\tilde{\mbf{w}}(\xi),
\end{equation}
with spatially $p$-periodic $\tilde{\mbf{w}}$ and spatial Floquet exponent
$\nu\in\C$. As usual, $\nu$ is only determined up to integer multiples
of $2\pi\rmi/p=\rmi k_{\rm wt}$. Explicitly,
\begin{equation}\label{e:bloch1d}
(\partial_\xi - \nu)^2 \tilde{\mbf{w}} + (\partial_\xi-\nu)\tilde{\mbf{w}} +
a(\xi)\tilde{\mbf{w}} = \lambda\tilde{\mbf{w}}
\end{equation}
Alternatively, we can work with $\mbf{w}$ directly and impose the Floquet
boundary condition
\[
\mbf{w}(p)=\rme^{-\nu p} \mbf{w}(0).
\]
The Floquet exponents $\nu$ play the role of the spatial wave numbers
in the dispersion relation (\ref{e:dr}). Looking for solutions with
neutral growth $\nu=\rmi k$, $k\in\R$, in analogy to the spectral
results of Propositions \ref{p:ivc}, \ref{p:fredi}, we obtain the
dispersion relation
\begin{equation}\label{e:drper}
d(\lambda,\rmi k) ={\rm det}\,(\Phi(\lambda)-\rme^{-\rmi k p})= 0,
\end{equation}
where $\Phi(\lambda)$ denotes the period map to the linear differential
equation (\ref{e-fl}). Note that the dispersion relation is invariant
under the Floquet shift $k\to k+ k_{\rm wt}$. The values of $k$ in the
dispersion relation (\ref{e:drper}) are referred to as {\em Bloch
wavenumbers}. The associated spectral values $\lambda^\ell(\rmi
k),\,\ell=1,\ldots,N,$ are the eigenvalues of the elliptic operator
${\cal L}_*$ on $L^2(\R)$. The eigenfunctions $u(\xi)$ are called {\em
Bloch waves} associated with the Bloch wavenumber $k$. We can also
define group velocities
\[
c^\ell_{\rm g}(k):=
\frac{\rmd \Im\lambda^\ell(\rmi k)}{\rmd k},
\]
in analogy to (\ref{e:cg}). Analogous statements to Propositions
\ref{p:ivc}, \ref{p:fredi} then hold for equations with asymptotically
periodic coefficients, with Fredholm borders $\Gamma^\ell$ defined
through (\ref{e:drper}).

\subsection{Instabilities of travelling waves}
\label{s:1.3}
In this section, we investigate  relative
equilibria which lose stability for the PDE~(\ref{e:rdsys1})
\[
u_t = D u_{xx} + f(u;\mu),
\]
on the unbounded real line $x\in\R$.  
Here, $\mu\in\R$ denotes a typical control parameter, driving the
instability, which we assume to occur at the value
$\mu=0$. Instabilities can be caused either by point spectrum or by
essential spectrum crossing the imaginary axis. Without striving for
completeness, we present four different cases of Hopf bifurcation from
travelling waves, here. Three of them are caused by essential spectrum
crossing the imaginary axis.

Fredholm indices play a most prominent role when solving the resulting
nonlinear equations, locally. Lyapunov-Schmidt reduction generally
provides reduced equations, which map the kernel to the cokernel. Because the
boundaries of the essential spectrum were characterized precisely by
the loss of Fredholm property, in Section~\ref{s:1.2}, we will rely on
spatial dynamics ideas, in the spirit of~\cite{Ki82,fischer84} to discuss these
latter bifurcations.

\subsubsection{Instability of a front  caused by point spectrum}

First consider a travelling wave $u=q_*(x-c_* t)$ with essential
spectrum of the linearization ${\cal L}_*$ strictly contained in the
open left half-plane, see (\ref{e:rdsys1}),(\ref{e:fb}). We assume that
there is a pair of simple, purely imaginary eigenvalues crossing the
imaginary axis. In addition, there necessarily is a zero eigenvalue
due to translation of the relative equilibrium $q_*$. This additional
eigenvalue prevents the straight-forward application of standard Hopf
bifurcation theorems. We choose to proceed by center manifold
reduction, instead. With a careful choice of cut-off functions, we can
construct a center manifold which is invariant under
translations~\cite{Henry,SSW1}. With an appropriate choice of
coordinates, we find a three-dimensional reduced system of ODEs on the
center manifold, which is of skew-product form
\begin{equation}
\dot v = h(v;\mu),\qquad \dot g = c(v;\mu).
\end{equation}
Here, $v\in \cal U\subset \C$, $g\in\R$, $h(0;0)=0$ and ${\rm
spec\,}h_v(0;0)=\{\pm\rmi\omega\}$. The (small) $v$-variable
parameterizes the Hopf eigenspace. The (global) variable $g$ in the
translation group $G=\R$ parameterizes the position of the front. Since
we can preserve the translation equivariance of the original
reaction-diffusion system (\ref{e:rdsys1}) through the center manifold
reduction, the Hopf bifurcation in the $v$-equation is independent of
the translation component $g$. As a result, we typically encounter a
generic Hopf bifurcation in the $v$-equation.  Unique periodic orbits
$v_*(t;\mu)$ arise super- or subcritically, depending only on the sign
of a certain cubic normal form coefficient. The $v$-dynamics enter the
$g$-equation, with the periodic solution $v_*(t;\mu)$ acting as a
time-periodic forcing. Phenomenologically, this leads to a
periodically oscillating speed $c=c(v_*(t;\mu);\mu)$ of the wave front
position. The periodic variable $v_*(t;\mu)$ indicates small periodic
shape fluctuations of the propagating wave. 

We refer to Section~\ref{s:meander} for a thorough discussion in the
more complicated, but analogous situation of meandering spiral wave
patterns. 

\subsubsection{The Turing instability}
\label{s:turing}
When the essential spectrum of the linearization ${\cal L}_*$ at a
front touches the imaginary axis, a naive finite-dimensional
Lyapunov-Schmidt reduction fails, due to the absence of the Fredholm
property. The dynamics, generated by a continuum of eigenmodes, cannot
be represented by a finite collection of ordinary differential
equations.  We illustrate the effects of critical essential spectrum
as opposed to critical point spectrum by means of an example: one of
the asymptotic states of a propagating front experiences a Turing
instability. We begin with some background on the nature of this
pattern forming mechanism, predicted by Turing in 1952~\cite{Tur}, and
observed, only in 1989, in laboratory experiments~\cite{firstturing}.

A {\em Turing instability} occurs at one of the asymptotic states
$q_\pm$ if the spectrum of ${\cal L}_\pm$ in the steady frame touches
the imaginary axis at $\lambda=0$, with spatially inhomogeneous
critical eigenfunction. To fix ideas, we assume that the Turing
instability occurs at $q_+$ and we set $q_+(\mu)\equiv 0$. We consider the
reaction-diffusion system (\ref{e:rdsys1}) in the steady frame, first
\[
u_t=Du_{xx}+f(u;\mu),\quad x\in\R,
\]
with linearization 
\[
{\cal L}_0(\mu)=Du_{xx}+f'(0;\mu)u.
\]
In the notation of Section~\ref{s:1.2}, one of the curves $\Gamma^\ell_+$
touches the imaginary axes with a zero eigenvalue $\lambda(\rmi
k_*)=0$, for nonzero wavenumber, $k_*\not=0$; see Figure~\ref{f:tur}.
\begin{figure}[!ht]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,2.9)%
\centerline{\includegraphics[scale=1.0]{Fig/tur-disp.eps}}
\end{picture}%
\caption[]{The critical curve $\Gamma^\ell_+\subset \C$, (a), and the
temporal eigenvalue $\lambda^\ell_+$ as a function of the wavenumber
$k$, (b), are shown at criticality.}
\label{f:tur} 
\end{figure}
We assume that $\lambda^\ell(k_*;\mu=0)=0$ is a simple root of the
dispersion relation (\ref{e:dr}), which crosses the imaginary axis with
nonvanishing speed. After a suitable scaling, we find
\begin{equation}
\label{e:drtur}
\lambda^\ell(k;\mu_0)=\mu-(k-k_*)^2 + \rmO((k-k_*)^3),
\end{equation}
for $k\sim k_*$.  Since ${\cal L}_0$ possesses real coefficients, the
curve $\Gamma^\ell_+$ is necessarily contained in the real axis close
to the origin.

It was Turing's original discovery that this instability can actually
occur in reaction-diffusion systems, although the spatially
homogenizing effect of diffusion suggests that the most unstable
eigenfunction ought to be spatially homogeneous.  In case of a Turing
instability, a spatially structured eigenfunction $u_*\cos(k_* x)$ is
amplified by the time evolution for the linearization of the
reaction-diffusion system (\ref{e:rdsys1}) about the critical state
$q_+=0$
\begin{equation}\label{e:rl}
u_t=Du_{xx} + f'(0;0)u.
\end{equation}
The resulting spatially periodic,  stationary pattern for the
nonlinear equation (\ref{e:rdsys1}) near $0$ is generally referred to
as a {\em Turing pattern} $T(x)$.

One possible theoretical approach to this instability imposes spatial
periodicity with period $L\sim 2\pi/k_*$ in the function space,
restricting the allowed values of the wavenumber of $k$ to multiples
of $2\pi/L$; see \cite{gostsc88}. Then the Turing bifurcation reduces to a
pitchfork bifurcation in the space of even functions with prescribed
period $L$, with reduced equation on a center manifold  given by
\begin{equation}\label{e:pf}
\dot z= (\mu-(k-k_*)^2) z + \beta(k;\mu) z^3 + \rmO(z^5)\in\R
.\end{equation}
We will assume throughout that $\beta(k_*;0)<0$, such that  the
bifurcating solutions are asymptotically stable --- in the space of
functions with the prescribed period $2\pi/k$. 

In a slightly different spirit, Turing instabilities on unbounded
domains, without imposing periodicity, have been described by a
partial differential equation instead of our single ordinary
differential equation. The dynamics of the reaction-diffusion system
can be approximated by a slow, long-wavelength modulation of
the complex amplitude of the critical eigenfunction:
\[
u(t,x)\sim \sqrt\mu\left(A(\mu t,\sqrt{\mu}x)\rme^{\rmi
k_* x}+\bar{A}(\mu t,\sqrt{\mu}x)\rme^{-\rmi k_* 
x}\right) u_*. 
\]
The time evolution of the complex amplitude
$A(\tilde{t},\tilde{x})\in\C$ is to leading order given by a
Ginzburg-Landau equation
\begin{equation}\label{e:GL}
A_{\tilde{t}}=  A_{\tilde{x} \tilde{x}} + A - A|A|^2,
\end{equation} 
on time scales
$\tilde{t}=\rmO(1)$.  In the
Ginzburg-Landau description (\ref{e:GL}), the stationary Turing
patterns are solutions of the form
$A(\tilde{x})=r(\tilde{k})\rme^{\rmi\tilde{k}\tilde{x}}$. Their
linearized stability can be explicitly calculated. In
$(\mu,\tilde{k})$-parameter space, only Turing patterns which are
sufficiently close to the critical wavenumber, $k=k_*$, or
$\tilde{k}=0$, are stable --- with respect to not necessarily
spatially periodic perturbations. The boundary of stability is
commonly referred to as the Eckhaus boundary; see Figure~\ref{f:eckh}.
\begin{figure}[!ht]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,3.5)%
\raisebox{-0.3cm}{\centerline{\includegraphics[scale=0.5]{Fig/tur-eckh.eps}}}
\end{picture}%
\caption[]{Existence and stability of Turing patterns depending on
spatial wavenumber $k$ and bifurcation parameter $k$.}
\label{f:eckh} 
\end{figure}
Turing patterns $T(x;\mu,k)$ exist for parameter values $\mu>\mu_{\rm
ex}(k-k_*)$, and are asymptotically stable for $\mu>\mu_{\rm
st}(k-k_*)\geq\mu_{\rm
ex}(k-k_*)$. The boundary curves can be computed to leading
order from the Ginzburg-Landau approximation~(\ref{e:GL}):
\[
\mu_{\rm ex}(\tilde k)=\tilde{k}^2 + \rmO(\tilde{k}^3),\quad \mu_{\rm
st}(\tilde k)=3\tilde k^2+ \rmO(\tilde k^3).
\] 
We refer to~\cite{CH,SchMie,MielkeSH,SSmodstab,Schneider} for physics
background and mathematical proofs in this context.


\subsubsection{Essential Hopf instability of a front}
\label{s:essh}
We are now going to describe a general bifurcation result for a front,
where one of the asymptotic states undergoes a Turing
instability. Reflecting the spatial variable $x$, if necessary, we may
fix $c_*>0$, that is, propagation towards $x=+\infty$. Then the
direction of propagation of the front distinguishes between the
asymptotic states $q_\pm$. We refer to $q_+$ as the state {\em ahead}
of the front, and to $q_-$ as the state which is {\em behind} the
front.

Whereas the Turing bifurcation is stationary in the steady frame, the
solution $u_*\exp{\rmi k_* x}$ to the linearized
equation~(\ref{e:rl}) becomes time-periodic $u_*\exp(\rmi k_* \xi +
\rmi \omega_* t)$, in a comoving frame with speed
$c_*=\omega_*/k_*>0$. Viewed in the comoving frame, the Turing
bifurcation therefore becomes a Hopf bifurcation. Typically, in a Hopf
bifurcation, we are interested in time-periodic solutions. More
specifically, we shall be interested in the existence and stability of
{\em modulated front solutions} to the reaction-diffusion
system~(\ref{e:rdsys1}):
\[
u(t,x)=q(t,x-ct),\quad \mbox{with } q(t,\xi)=q(t+P,\xi), \mbox{ for
all }t,\xi\in\R,
\]
for some temporal period $P$, and an appropriate wave speed $c\sim c_*$.

Setting up a typical bifurcation result usually requires {\em
existence} of a primary solution branch, {\em minimal critical
spectrum}, and assumptions on {\em nonlinear terms} in the Taylor
expansion. We are now going to make these assumptions precise for our
set-up. Recall that $x$ always refers to the spatial
coordinate in the steady frame, whereas $\xi$ always refers to some
comoving frame. 

At criticality $\mu=0$, we assume {\em existence} of a uniformly
translating front solution, $q(x-c_*t)\to q_\pm$ for $x\to\pm\infty$
for some strictly positive speed of propagation $c_*>0$.

The {\em critical spectrum} is assumed to generate a Turing
instability at one of the asymptotic states, either ahead of the
front, at $q_+$, or behind the front, at $q_-$.  In the comoving frame
$\xi=x-c_* t $, this Turing instability corresponds to a spectral
curve $\Gamma_+$ (or $\Gamma_-$, respectively) of the linearization
about the front ${\cal L}_*$, (\ref{e:rdsys1l}) which touches the
imaginary axis. The tangency at $\lambda(\rmi k_*)=\rmi\omega_*\not=0$
is quadratic from the left. From the dispersion relation (\ref{e:dr}),
we find $\omega_*=c_* k_*$. Increasing the parameter $\mu$ through
zero, we assume that $\Gamma_+$ (or $\Gamma_-$, respectively) crosses
the imaginary axis with nonzero speed.

{\em Minimal} spectrum at bifurcation, here, refers to the spectrum of
the front in a comoving frame: we assume that the critical spectrum
only consists of the part related to the Turing instability and the
simple eigenvalue $\lambda=0$, induced by translation. More precisely,
we assume that $\lambda(\rmi k_*)=\pm\rmi\omega_*$ are the only roots
of the dispersion relation~(\ref{e:dr}) in the comoving frame $c=c_*$
on the imaginary axis $\lambda\in\rmi\R$, and
$\lambda=\pm\rmi\omega_*$ are simple roots. Moreover, we require
absence of critical point spectrum on the imaginary axis: in an
exponentially weighted space $L_\eta^2(\R)$ with norm
\[
\|u\|^2_{L_\eta^2}:=\int_\R|\rme^{\eta\xi}u(\xi)|^2 <\infty
,\]
we assume that  
\[
{\rm spec\,}_{L_\eta^2}{\cal L}_*\cap \rmi\R =\{0\},
\]
is algebraically simple, for all $\eta>0$ sufficiently small. Note
that compared to Lemma~\ref{l:cgweight}, we used exponential weights
$\eta_+=\eta_-:=\eta$. 

{\em Nonlinear terms} for the Turing bifurcation in a neighborhood of
the asymptotic state $q_+$ (or $q_-$, respectively) are most easily
computed in the steady frame, restricting to even, $2\pi/k$-periodic
functions. As mentioned above, the instability then induces a
pitchfork bifurcation on a one-dimensional center manifold, with
Taylor expansion
\[
\dot z=\lambda(\rmi k;\mu) z + \beta(k;\mu) z^3 + \rmO(z^5).
\]
We assume $\beta(k_*;0)<0$ such that the bifurcating Turing patterns
are stable within this class of periodic functions. From elementary
bifurcation theory, we find for each $k\sim k_*$ and $\mu>0$, $\mu\sim
0$, a  periodic Turing pattern
$T(x;\mu,k)$, which is time-independent in the original frame $x$ and unique up to translation in $x$.


\begin{Theorem}\label{t:1}\cite{SSfront}
Assume existence of a front solution $q_*(x-c_*t)$, $c_*>0$, with
minimal, critical essential spectrum causing a supercritical Turing
instability, as described above.
\begin{itemize}
\item[(I)] Assume the Turing instability occurs ahead of the front, at
$q_+$, only. Then, for each Turing pattern $T(x;\mu,k)$ with $(k,\mu)$
near $(k_*,0)$, there exists a modulated front solution of the
reaction-diffusion system~(\ref{e:rdsys1}), $q(t,x-c(\mu,k)t;\mu,k)$,
invading the Turing pattern
\[
q(t,\xi;\mu,k)\to
T(\xi+c(\mu,k)t;\mu,k),\quad \mbox{ for } \xi\to\infty,
\]
uniformly in time $t$. The modulated front is $\sqrt\mu$-close to the
primary front $q_*$, uniformly in $\xi$ and $t$, and is
$P(\mu,k)$-periodic in its first argument, with $P(0,k_*)=2\pi
/(c_*k_*)$ and $c(0,k_*)=c_*$. See Figure~\ref{f:fig-ta} for a
schematic plot. The bifurcating modulated front is unique in the class
of modulated fronts, close to the primary front $q_*(\xi)$, invading a
fixed Turing pattern, up to spatial and temporal translations.

If the bifurcating Turing pattern is  stable, that is, if it lies
inside the Eckhaus boundary, then the  modulated front is spectrally
stable.

\item[(II)] Assume the Turing instability occurs behind the front, at
$q_-$, only. Then for any parameter value $\mu\sim 0$, any speed $c\sim c_*$,
any wavenumber $k\sim k_*$, and any temporal period $P\sim
2\pi/(c_*k_*)$, there does {\em not} exist a modulated front solution
$q(t,x-ct)$ to (\ref{e:rdsys1}), which is $P$-periodic in the first
argument, and which leaves a Turing pattern behind
\[
q(t,\xi)\to
T(\xi+ct;\mu,k),\quad \mbox{ for } \xi\to-\infty
;\]
see Figure~\ref{f:fig-tb}.
\end{itemize}
\end{Theorem}

\begin{figure}[!ht] 
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,3.5)% 
\centerline{\includegraphics[scale=0.7]{Fig/fig-ta.eps}}
\end{picture}% 
\caption[]{A modulated front  invading a Turing pattern, case I}  
\label{f:fig-ta}  
\end{figure} 
 
\begin{figure}[!ht] 
\setlength{\unitlength}{1cm}% 
\begin{picture}(13.5,3.5)% 
\centerline{\includegraphics[scale=0.7]{Fig/fig-tb.eps}}
\end{picture}% 
\caption[]{Non-existence of modulated fronts leaving a Turing pattern behind, case II.} 
\label{f:fig-tb}  
\end{figure} 

Stability in the theorem refers to the spectrum of the temporal period
map of the reaction-diffusion system, linearized at the
(time-periodic) modulated front in a comoving coordinate frame.

If the front moves to the left, $c_*<0$, we may reflect $x$ and find
bifurcating modulated waves for an instability at $q_-$. In this
sense, bifurcation, case I, occurs whenever the front is 
invading the Turing unstable state, and bifurcation failure, case
II, occurs when the front leaves the Turing unstable state behind.

We emphasize that bifurcation failure, case II, is in contrast to finite-dimensional
bifurcation theory, where generically periodic solutions arise in Hopf
bifurcation.


\subsubsection{Instability of a pulse caused by the essential spectrum}

If $q_+=q_-=0$, bifurcation is in conflict with bifurcation failure in
Theorem~\ref{t:1}, above. The Turing instability then occurs ahead of
the front and behind the front, simultaneously. This scenario
naturally arises if we consider pulses
\[
0 \not\equiv q_*(\xi)\to 0,\ {\rm for} \ |\xi|\to\infty
.\]

\begin{Theorem}\label{t:2}\cite{SSmodpulse}
Assume existence of a pulse solution $q_*(x-c_*t)$, with minimal,
critical essential spectrum causing a supercritical Turing instability
as described in Section~\ref{s:essh}.

Then, for each Turing pattern $T(x;\mu,k)$ with $(\mu,k)\sim (0,k_*)$,
there exists a modulated pulse solution of the reaction-diffusion
system~(\ref{e:rdsys1}), $q(t,x-c(\mu,k)t;\mu,k)$, travelling through
the Turing pattern. More precisely, there are $(\mu,k)$-independent
constants $C,\eta,\eta'>0$, and phases $\theta_\pm(\mu,k)$ such that
\begin{eqnarray*}
|q(t,\xi)-
T(\xi+ct + \theta_+)|&\leq & C\rme^{-\eta|\xi|},\mbox{ for all }\xi>0,\\
|q(t,\xi)-
T(\xi+ct + \theta_-)|&\leq & C\rme^{-\eta'\mu|\xi|},\mbox{ for all }\xi<0.
\end{eqnarray*}
We have suppressed dependence of $q$, $T$, $c$, and $\theta_\pm$ on
$(\mu,k)$. The modulated pulse is $\sqrt\mu$-close to the primary
pulse, uniformly in $\xi$ and $t$, and periodic in its first argument $t$,
with period $P=P(\mu,k)$, $P(0,k_*)=2\pi /(c_*k_*)$ and
$c(0,k_*)=c_*$. See Figure~\ref{f:fig-tab} for a schematic plot. The
bifurcating modulated pulse is unique in the class of modulated
pulses, $(x,t)$-uniformly close to the primary pulse $q_*(\xi)$,
moving through a fixed Turing pattern, up to spatial and temporal
translations.

If the bifurcating Turing pattern is  stable, that is, if it lies
inside the Eckhaus boundary, then the  modulated pulse is spectrally
stable.
\end{Theorem}
Again, spectral stability refers to the linearized temporal period-$P$ map along
the time-periodic modulated pulse.
\begin{figure}[!ht] 
\setlength{\unitlength}{1cm}% 
\vspace*{0.6cm}
\begin{picture}(13.5,3.5)% 
\centerline{\includegraphics[scale=0.7]{Fig/fig-p.eps}} 
\end{picture}% 
\caption[]{A modulated pulse moving through a Turing pattern}   
\label{f:fig-tab}  
\end{figure} 

Summarizing, we find bifurcation of time-periodic solutions in this
case of simultaneous instability ahead and behind a pulse, just like
in the case of fronts invading the Turing pattern. Reminiscent of the
nonexistence in case of an instability behind the front is the
recovery zone of length $1/\mu$ behind the pulse, marked by the slow
convergence to the Turing patterns for $\xi\to -\infty$.

\subsubsection{Fredholm indices and essential instabilities}

There are several ways of understanding Theorems~\ref{t:1} and
\ref{t:2}. We first appeal to intuition.

Consider bifurcation of fronts case I, first. Before the instability,
$\mu<0$, the homogeneous state $q_-$ on the left invades the stable
state $q_+$ on the right by means of the front moving with speed
$c\sim c_*>0$. For $\mu>0$ the homogeneous state ahead of the front
has become unstable and Turing patterns have formed. Intuition
suggests that the front would invade these Turing patterns close to
the homogeneous state $q_+$, which was invaded for $\mu<0$.

Consider bifurcation failure, case II, next, with $\mu>0$. We expect
the dynamics near $q_-$ behind the front to be described approximately
by the Ginzburg-Landau equation (\ref{e:GL}). In this approximation,
there exist travelling waves, where a Turing pattern invades the
unstable homogeneous background $A=0$. For example, we may consider
the Turing pattern $A(x)\equiv 1$ constant. The spread of $A\equiv 1$
into the unstable background $A\equiv 0$ (alias $q_-$), is then
described by the second order travelling wave equation
\[
A''+\tilde{c}A'+A-A^3=0,
\]
for solutions $A=A(\tilde{x}-\tilde{c}\tilde{t})$ of
(\ref{e:GL}). Monotone heteroclinic orbits, connecting $A=1$ to $A=0$
exist for all speeds $\tilde{c}\geq 2$. In the unscaled coordinates
$t,x$, the speed $\tilde{c}=2$ corresponds to a speed $c_{\rm
Turing}=\rmO(\sqrt{\mu})$. The heteroclinic represents a small front
of a Turing pattern to the left invading the unstable state $q_-$, to
the right. At present, there is no proof, that Turing patterns
actually spread with this (minimal) speed into the unstable
homogeneous background state; see, however,~\cite{CoEc} for evidence
in this direction. Now observe that in this picture, the speed of a
front between a Turing pattern and the homogeneous unstable state
$q_-$ is much slower than the speed of the primary front. Imagine an
initial condition consisting of the Turing pattern on the left,
followed to the right by the unstable state $q_-$, then the front, and
finally the stable homogeneous state $q_+$ on the right. Then the
interface between the Turing pattern and the unstable state $q_-$
moves much slower than the primary front. This leads lead to a wedge
opening between the two fronts; see Figure~\ref{f:wedge}.

\begin{figure}[!h] 
\setlength{\unitlength}{1.0in}%
\begin{picture}(0,0)%
{\raisebox{-0.3cm}{\centerline{\includegraphics[scale=0.8]{Fig/wedge.eps}}}}
\end{picture}%
\begin{picture}(5.5,2.0)
\put(5.,.8){\vector( 0, 1){0.6}}
\put(5.,.8){\vector(-1, 0){0.6}}
\put(5.,.8){\vector(-1,-3){0.2}}
{\small
\put(1.6,1.75){$q_+$}
\put(2.35,.10){$q_-$}
\put(3.70,1.35){$T(x)$}

\put(4.50,.85){$x$}
\put(4.88,.20){$t$}
\put(4.81,1.20){$u$}
}
\end{picture}
\caption[]{The wedge of an unstable state formed between a small amplitude front of Turing patterns $T(x)$ and a large amplitude front connecting $q_+$ and $q_-$.}   
\label{f:wedge}  
\end{figure} 

Another explanation would take a functional analytic point of view,
still formal, and allude to the results on Fredholm properties of the linearization, Proposition~\ref{p:ivc}. 

Consider case I, first. Observe that the Fredholm index of the
linearization ${\cal L}_*-\rmi\omega_*$ along the front in a comoving
frame changes at the bifurcation point $\mu=0$. The group velocity
associated with the Turing instability in a steady frame is zero,
since $\lambda(\rmi k)$ is real for all $k\sim k_*$. In a comoving
frame of speed $c_*>0$, the group velocity is negative. Therefore, by
Proposition~\ref{p:ivc}, the Fredholm index changes from $0$ to $+1$
when crossing the critical spectral curve $\lambda(\rmi k)$ from the
unstable to the stable complex half plane; see Figure~\ref{f:spectu}.

\begin{figure}[!h] 
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,3.)%
\centerline{\includegraphics[scale=1.0]{Fig/spec-essinst.eps}}
\end{picture}%
\caption[]{Spectra, Fredholm indices, and oriented critical spectral
curves of the linearization about a front, when a Turing instability
occurs ahead of the front (a) or behind the front (b).}
\label{f:spectu}  
\end{figure} 
On the linear level, this index computation suggests that after the
instability, $\mu>0$, the Fredholm index of the linearization at
$\rmi\omega_*$ is $+1$, such that the linearization possesses a
one-dimensional kernel, and we should be able to find a family of
solutions, parameterized by the wavenumber $k$.

In case II, the analogous argument shows that the Fredholm index is
$-1$ after the instability, $\mu>0$, since, again, the associated
group velocity is negative, but now points away from the interface;
see Proposition~\ref{p:ivc}. We therefore do not expect to find any
periodic solution since the cokernel is at least one-dimensional.

However, this reasoning remains formal, since right at the moment of
bifurcation, $\mu=0$, the linearization is not Fredholm and
Lyapunov-Schmidt reduction does not apply. In fact, the reasoning
fails in case of Theorem~\ref{t:2}. The Turing instability occurs
simultaneously ahead and behind the front. Therefore, two identical
spectral curves touch the imaginary axis simultaneously, both with
negative group velocity. One of the curves is associated with the
linearization at $q_+$, one with $q_-$. Again, by
Proposition~\ref{p:ivc}, the Fredholm index to the left and to the
right of these two identical critical curves is zero.  Still, a family
of solutions parameterized by the wavenumber $k$ bifurcates, just like
in the case of Fredholm index $+1$; see Theorem~\ref{t:2}.

\subsubsection{Spatial dynamics and essential instabilities} 
\label{s:sdei} 
We now outline our strategy for the proof of Theorems~\ref{t:1} and
~\ref{t:2}. The proofs rely on a spatial dynamics formulation of the
parabolic equation~(\ref{e:rdsys1l}) in the spirit
of~\cite{Ki82}. Consider the system
\begin{eqnarray}\label{e:spatdyn} 
U_\xi & = & V\nonumber\\ V_\xi & = & D^{-1}(-cV + \omega \partial_t U
- F(U;\mu)),
\end{eqnarray} 
in a space $(U,V)\in X=H^{1/2}(S^1)\times L^2(S^1)$ of functions which
are $2\pi$-periodic in time $t\in S^1$. The wave speed $c$ and the
temporal period $P=2\pi/\omega$ enter as additional parameters into
the equation. We view system (\ref{e:spatdyn}) as an abstract
differential equation for $\mbf{U}=(U,V)(\cdot)$
\begin{equation}\label{e:a}
\mbf{U}_\xi = \mbf{F}(\mbf{U};\omega,c,\mu)\in X.
\end{equation}
The goal is to find heteroclinic orbits to this abstract differential
equation on a Hilbert space. Indeed, modulated fronts are
periodic in time $t$ for an appropriate wave speed $c$. In spatial ``time''
$\xi$, they converge to the asymptotic states $q_\pm$ or the Turing
patterns. The Turing patterns are periodic in time $t$ and spatial
``time'' $\xi$, in the comoving frame. They arise out of a Hopf
bifurcation from the equilibrium which undergoes a Turing
instability. The essential instability therefore reduces to a global
heteroclinic bifurcation, where a heteroclinic orbit connects two
equilibria, one of which undergoes a Hopf bifurcation. 

Before we proceed to analyze this heteroclinic bifurcation, we want to
point out the ill-posed character of (\ref{e:a}). The initial value
problem to this equation is ill-posed, even in the example of the
simple heat equation
\begin{equation}\label{e:heat}
U_\xi = V,\qquad V_\xi = \partial_t U.
\end{equation}
Indeed, we have solutions $U(\xi,t)=\rme^{\rmi\ell t + \nu_\ell \xi}$
to (\ref{e:heat}), with $\nu_\ell=\pm\sqrt{\rmi \ell}$, for any
$\ell\in\Z$. In particular, there is no a priori bound on the
exponential growth of solutions, neither in forward, nor in backward
spatial ``time'' $\xi$. However, solutions to the initial value
problem are unique, provided they exist~\cite{SSmathnach,Chen}. The
situation is reminiscent of the elliptic boundary-value problem
$u_{\xi\xi}+u_{yy}=f(u)$, posed on unbounded cylinders $x\in\R$, $y\in
(0,1);$ see for example ~\cite{Ki82,fischer84,mielke94,fiscvi98}.

Still, we can formally linearize (\ref{e:a}) at equilibria and compute
the spectrum of the linearized operator. It then turns out that, just
as in the case of the heat equation (\ref{e:heat}), most of the
eigenvalues are bounded away from the imaginary axis. It is therefore
possible to define stable, unstable, and center manifolds in certain
regions of the phase space $X$.  In particular, given a solution
$\mbf{U}_*(\xi)$ to the nonlinear equation, we can construct stable
and unstable manifolds in the vicinity of this solution if the
linearized equation
\[
\mbf{W}_\xi = \mbf{F}'(\mbf{U}_*(\xi);\omega,c,\mu)\mbf{W}
\]
possesses an {\em exponential dichotomy}; see~\cite{PSS,SSmathnach}
for a definition. Existence of exponential dichotomies on $\R_+$
is guaranteed, if, $\mbf{U}_*(\xi)\to\mbf{U}_{*,+}$ for
$\xi\to\infty$ and the linearized equation about $\mbf{U}_{*,+}$
does not possess solutions of the form $\mbf{W}(t)\rme^{\rmi
k\xi}$ for any $k\in\R$; see~\cite{PSS,SSmathnach}. Similarly, center,
center-unstable, and center-stable manifolds can be constructed, if
the weighted linearized equation 
\[
\mbf{W}_\xi = \mbf{F}'(\mbf{U}_*(\xi);\omega,c,\mu)\mbf{W}
-\eta_+ \mbf{W}
,\]
possesses an exponential dichotomy; see again~\cite{PSS,SSmathnach}.

The main advantage of our spatial dynamics formulation is that the
independent variable $t$ lives on a compact domain, the circle. Time
shift provides an $S^1$-equivariance in phase space ${\mbf U}\in X$
\begin{equation}\label{e:sgr}
\mbf{U}(\cdot)\mapsto \mbf{U}(\cdot + \theta)\in X,\quad
\theta\in S^1;
\end{equation}
see also~\cite{vander82} for the role of $S^1$-symmetry in Hopf
bifurcation. 
The subspace of functions $(U,V)$ which are fixed under this
action of the circle group consists of the time-independent
functions. In this subspace, the  abstract 
differential equation~(\ref{e:a}) reduces to the  usual travelling
wave ODE
\[
U_\xi=V,\quad V_\xi = D^{-1} (-cV - F(U;\mu)).
\]
For $\mu=0$, there is a heteroclinic orbit
$\mbf{U}=Q(\xi)=(q(\xi),q'(\xi))$ to this differential equation,
converging to the ``equilibria'' $Q_\pm=(q_\pm,0)$. Recall again, that the
term ``equilibria'', here, refers to spatial dynamics: ``equilibria'' are
spatially homogeneous solutions.

Consider first the instability ahead of the front, case I in
Theorem~\ref{t:1}. We fix the temporal frequency to
$\omega_*=c_*/k_*$. The Turing instability turns into a Hopf
bifurcation, reflecting spatial periodicity, with (spatial)
eigenvalues $\pm\rmi k_*$ from the dispersion relation
(\ref{e:drtur}).  Turing patterns lie in a small neighborhood of
$Q_+$, but outside of the subspace of time-independent functions.
Simplicity of the critical spectral curve, which induces the Turing
instability guarantees that the linearization at $Q_+$ possesses
precisely two simple imaginary eigenvalues $\nu_{\rm H}=\rmi k_*$ and
$\bar{\nu}_{\rm H}$, constituting the center eigenspace. After
reduction to a 2-dimensional center manifold in a neighborhood of the
equilibrium $Q_+$, we find a periodic orbit arising through the Hopf
bifurcation. Temporal time shift acts nontrivially on the
center-manifold such that the periodic orbits actually are relative
equilibria, with spatial time evolution given by the temporal
time-shift symmetry. 

We next argue that the Turing patterns are unstable with respect to
the spatial $\xi$-dynamics inside the spatial center manifold. Since
$\lambda$ is real in the steady frame, the group velocity associated
with the critical spectral curve in the comoving frame is negative;
see~(\ref{e:cg}) and the following remark. By Lemma~\ref{l:cgweight},
the temporal time-$t$-instability disappears in a space of functions
with weight $\rme^{\eta \xi}$, $\eta>0$ small; see
Lemma~\ref{l:cgweight}. In our spatial dynamics picture, this weight
shifts the spatial eigenvalues $\nu$ of the linearization at the
equilibrium $Q_+$, adding $\eta>0$. The effect is that the spatial
eigenvalue $\nu_{\rm H}(\mu)$ stays on the side of the imaginary axis,
where it was located for $\mu<0$, suppressing neutral spatial
eigenvalues and failure of the Fredholm property throughout the
bifurcation. This is only possible, if $\Re\nu_{\rm H}(\mu)<0$ for
$\mu>0$, that is, if the equilibrium $Q_+$ is stable inside the center
manifold, after bifurcation. By exchange of stability, the periodic
solution is therefore unstable inside the center manifold.

We turn to the global bifurcation along the heteroclinic orbit
$Q(\xi)$, next; compare also Figure~\ref{f:fig-fa}. We argue with the
spatial $\xi$-dynamics naively, using objects like
(infinite-dimensional) stable and unstable manifolds of the
``equilibria'' $Q_\pm$ freely, as is usual for ODEs;
see~\cite{SSmodpulse,SSfront} for a technical justification. Just before
bifurcation, $\mu<0$, the only bounded solution of
equation~(\ref{e:spatdyn}), linearized along the front $Q(\xi)$, is
$Q'(\xi)$. Indeed, any other bounded solution would correspond to a
time-periodic solution of the linearized parabolic equation and hence
would yield an eigenvalue $2\pi\rmi\ell/\omega$ of the linearization
${\cal L}_*$, for some $\ell\in\Z$. Therefore, the intersection of the
global stable manifold of $Q_+$ and the global unstable manifold of
$Q_-$ consists of precisely the heteroclinic $Q(\xi)$. Upon varying
$c$, stable and unstable manifold cross with nonzero speed, since the
zero eigenvalue was assumed to be simple. This {\em transverse
crossing} is best viewed in the extended phase space, where the
equation~(\ref{e:spatdyn}) is augmented with an equation for the
parameter, $\mu_\xi=0$. In this extended phase space, transverse
crossing can be defined as a transverse intersection of the
$\mu$-family of stable manifolds with the $\mu$-family of unstable
manifolds. Through the bifurcation point $\mu=0$, the stable manifold
of $Q_+$ can be continued smoothly as the strong stable manifold of
$Q_+$. Indeed, considering the strong stable manifold eliminates the
additional weakly stable center direction, which appears at the
bifurcation point. Transverse crossing persists as transverse crossing
of the strong stable manifold of $Q_+$ and the unstable manifold of
$Q_-$, throughout the bifurcation $\mu\sim 0$. The central observation
now is that, by continuity of the stable fibration of the center
manifold, the strong stable manifold of $Q_+$ is close to any of the
strong stable fibers of the periodic orbit in the center manifold!
Therefore, the strong stable fibers of the periodic orbit cross
the unstable manifold of $Q_-$ transversely, which yields the desired
heteroclinic orbit. If we choose a different strong stable fiber of
the periodic orbit, we find a different heteroclinic
connection. However, any of those connections can be obtained from
the first heteroclinic through the time-$t$-shift symmetry. Uniqueness
of the heteroclinic implies uniqueness of the modulated front in the
class of functions with prescribed temporal frequency
$\omega$. Varying the temporal time-$t$-frequency $\omega$ amounts to
a different choice of the wavenumber of the Turing pattern through the
relation $k=c/\omega$. In other words, $\omega$ selects the spatial
wavenumber of the Turing pattern that we choose to be placed ahead of
the front.

\begin{figure}[!ht] 
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,6.)% 
\centerline{\includegraphics[scale=0.7]{Fig/fig-fa.eps}}
\end{picture}%
\caption[]{The heteroclinic bifurcation creating a modulated front
which invades a Turing pattern}
\label{f:fig-fa}
\end{figure}

The picture for the instability behind the front, bifurcation failure
case II, is very similar; compare Figure~\ref{f:fig-fb}. The local
bifurcation near $Q_-$ is now the same as the one discussed for $Q_+$
above. Again, we find a periodic orbit, unstable for the spatial
$\xi$-dynamics inside the center-manifold, representing Turing
patterns behind the front, in the center manifold close to
$Q_-$. Before bifurcation, $\mu<0$, the unstable manifold of $Q_-$ and
the stable manifold of $Q_+$ cross transversely. This time however, we
have to continue the unstable manifold of $Q_-$ through the
bifurcation point $\mu=0$. Since the Hopf eigenvalue $\nu_{\rm H}$ is
unstable before the bifurcation, $\mu<0$, the unstable manifold of
$Q_-$ continues smoothly to the center-unstable manifold of $Q_-$
after bifurcation, $\mu>0$.  Transversality yields a unique
intersection of the center-unstable manifold of $Q_-$ with the stable
manifold of $Q_+$ after bifurcation. However, this unique
intersection, is already present in the $2N$-dimensional subspace of
time-independent solutions: it is given by the trivial continuation of
the robust primary front. This proves non-existence of modulated
waves, leaving a Turing pattern behind, in the case of an instability
behind the front.

\begin{figure}[!ht] 
\setlength{\unitlength}{1cm}% 
\begin{picture}(13.5,5.5)% 
\centerline{\includegraphics[scale=0.7]{Fig/fig-fb.eps}}
\end{picture}% 
\caption[]{The heteroclinic bifurcation failure of a  front
leaving a Turing pattern behind }   
\label{f:fig-fb}  
\end{figure} 

For a pulse, we combine the two arguments above; see also
Figure~\ref{f:fig-p}. Throughout the bifurcation, the strong stable
manifold of the origin crosses the center-unstable manifold of the
origin transversely, along $Q(\xi)$. Again by continuity of the
strong stable fibration of the center manifold the strong stable
fiber of the periodic orbit, representing the Turing pattern ahead of
the front, is close to the strong stable manifold of the origin.  This
implies transverse heteroclinic crossing of this strong stable fiber
with the center-unstable manifold of the origin, at an
$\rmO(\mu)$-distance of the periodic orbit. As a consequence, the
intersection is not contained in the stable manifold of the origin,
which is distance $\rmO(\sqrt\mu)$ from the periodic Turing orbit. In
backward spatial time $\xi\to -\infty$, the homoclinic orbit slowly
approaches the periodic orbit: it creeps along the center manifold
with rate $\rmO(\exp(\eta'\mu\xi))$, as given by the linearization of
the periodic orbit inside the center manifold.

\begin{figure}[!ht] 
\setlength{\unitlength}{1cm}% 
\begin{picture}(13.5,5.5)% 
\centerline{\includegraphics[scale=0.7]{Fig/fig-fp.eps}} 
\end{picture}%
\caption[]{The homoclinic bifurcation creating a modulated pulse
travelling through a Turing pattern}  
\label{f:fig-p} 
\end{figure}

For the technically more involved stability considerations, we refer
to~\cite{SSfront,SSmodstab,SSmathnach}.

\subsection{From unbounded to large domains: absolute versus essential
spectra}
\label{s:2}

In view of the bifurcation results in Theorems~\ref{t:1} and
\ref{t:2}, we might try to describe the instability of fronts and
pulses in a large, but bounded, comoving domain. Essential spectra
disappear since the linearized operator ${\cal L}_*$ from
(\ref{e:rdsys1l}) possesses a compact resolvent on any finite
domain. Therefore any instability in a bounded domain is necessarily
due to point spectrum crossing the imaginary axis. In case of a pair
of complex conjugate eigenvalues crossing the axis, a general Hopf
bifurcation theorem guarantees bifurcation of periodic orbits ---
under mild nondegeneracy conditions. In consequence, there is an
obvious difficulty trying to approximate instabilities in unbounded
domains by instabilities in bounded domains.  In particular the
bifurcation failure in case II of Theorem~\ref{t:1}, cannot be
reproduced in any approximating bounded domain.

For a better understanding of this difficulty, we return to the simple
example of the advection-diffusion problem
\begin{equation}\label{e:cdd}
w_t=w_{\xi\xi} + w_\xi + \mu w =:{\cal L}_\mu w.
\end{equation}
Compared to~(\ref{e:cd}), we have added a constant linear driving term
$\mu w$. In the unbounded domain $x\in\R$, the instability threshold
is $\mu=0$, where the essential spectrum,
\[
{\rm spec}_{\rm ess}\,{\cal L}_\mu=\{\lambda=-k^2-\rmi k + \mu;\; k\in\R\}
,\]
crosses the imaginary axis. 

Truncating to $x\in 
(-L,L)$ and imposing periodic boundary conditions causes the curves of
essential spectra to break up into point spectrum. The eigenvalues are
still located in a neighborhood of the curve of essential spectrum, and
can be computed explicitly:
\[
{\rm spec}_{{\rm per},(-L,L)}\,{\cal
L}_\mu=\{\lambda_\ell=-\left(\frac{\pi\ell}{2L}\right)^2 +
\rmi\frac{\pi\ell}{2L} +\mu;\; \ell\in\Z\}.
\] 
On bounded subsets of the complex plane, ${\rm spec}_{{\rm
per},(-L,L)}\,{\cal L}_\mu$ converges to the essential spectrum ${\rm
spec}_{\rm ess}\,{\cal L}_\mu$, in the
symmetric Hausdorff distance, for $L\to\infty$.

Imposing separated boundary conditions, for example Dirichlet boundary
conditions $u(-L) = u(L) = 0$, the spectrum of ${\cal L}_\mu$ again
consists of point spectrum. However, the point spectrum is now located on
the real axis, with eigenvalues given explicitly by
\[
{\rm spec}_{{\rm Dir},(-L,L)}\,{\cal
L}_\mu=\{\lambda_\ell=-\left(\frac{\pi\ell}{2L}\right)^2+ \mu -\frac 1
4;\; \ell\in\Z\}.
\] 
The eigenfunctions are exponentially localized at the left boundary of
the interval $(-L,L)$. On bounded subsets of the complex plane, the
point spectrum converges to $(-\infty,\mu-\frac{1}{4}]$ in the
symmetric Hausdorff distance, for $L\to\infty$. We will later identify
this half line as the {\em absolute spectrum} of the operator ${\cal
L}_\mu$, posed on $x\in\R$.

Note that the half line $\lambda\leq \mu-1/4$ is not part of the
spectrum of ${\cal L}_\mu$ on $x\in\R$, but merely arises as a
set-wise limit of the spectrum if we truncate to finite intervals and
let $L$ tend to infinity. It is not difficult, but tedious, to check that
almost all separated mixed boundary conditions induce a similar
limiting behavior of the spectrum on large intervals $(-L,L)$, as the
length of the interval tends to infinity. For example, in the case of
Neumann boundary conditions, the spectrum converges
\[
{\rm spec}_{(-L,L),{\rm sep}}\,{\cal
L}_\mu\longrightarrow (-\infty,\mu-\frac{1}{4}]\cup \{\mu\}
\quad\mbox{ for } L\to\infty,
\]
on bounded subsets of the complex plane $\C$, in the symmetric
Hausdorff distance.  It is an interesting exercise to follow the
eigenvalue $\lambda=\mu$ through a homotopy from Neumann to Dirichlet
through mixed boundary conditions.

For Dirichlet boundary conditions and $0<\mu<\frac{1}{4}$, the system
is unstable on the real axis $x\in\R$, whereas it is stable under separated
boundary conditions, for any size of the domain. In this
parameter regime, we call the system
{\em convectively unstable}. Indeed, the instability on the unbounded
domain is convective in nature: starting with initial conditions
$u_0(x)$ having compact support on $\R$, the absolute value
$|u(t,x_0)|$ converges to zero for every fixed point $x_0$ on the real
axis, although the norm $\|u(t,\cdot)\|_{L^2}$ grows
exponentially. The pointwise decay can be extracted from the explicit
representation of solutions by the Green's function for~(\ref{e:cdd}).

This general phenomenon of pointwise decay of solutions, which
nevertheless grow in norm, was first emphasized in the context of
plasma physics and fluid flow instability~\cite{Landau,Briggs}. We
will now study the behavior of spectra of pulses or fronts in
reaction-diffusion systems, in the limit $L\to\infty$;
see~\cite{SStrunc} for a more general result.

Consider the linear reaction-advection-diffusion system
\begin{equation}\label{e:rcd}
{\cal L}u=Du_{\xi\xi}+ c u_\xi + a(\xi)u
,\end{equation}
where the $N\times N$-matrix $a(\xi)$ converges exponentially to $a_\pm$
for $\xi\to\pm\infty$. At $\xi=\pm\infty$, we encounter the asymptotic
linearized operators  
\[
{\cal L}_\pm u=Du_{\xi\xi}+ c u_\xi + a_\pm u
.\]
Associated with ${\cal L}_\pm$ are the dispersion relations 
\[
d_\pm(\lambda,\nu)={\rm det}\,(D\nu^2 - c\nu + a_\pm - \lambda)=0,
\]
which are polynomials of degree $2N$ in $\nu$. Let
$\nu^j_\pm(\lambda)$, $j=1,\ldots,2N$, denote the complex roots,
ordered by decreasing real part
\[
\Re\nu^1_+\geq \ldots\geq \Re\nu^{2N}_+,\quad \Re\nu^1_- \geq \ldots
\geq\Re\nu^{2N}_- 
.\]
Note that this numbering  might depend discontinuously on the spectral
parameter $\lambda$. 

In the homoclinic case $a_+=a_-$, the {\em essential spectrum}, ${\rm
spec\,}_{\rm ess}({\cal L})$, is given by the set of $\lambda$ such
that $\Re\nu_+^j(\lambda)=0$ or $\Re\nu_-^j(\lambda)=0$, for some $j$;
see Proposition~\ref{p:ivc}.

In the general case of a pulse or a front, we define the {\em absolute
spectrum}, ${\rm spec\,}_{\rm abs}({\cal L})$, as the set of
$\lambda\in\C$ such that $\Re\nu_+^N(\lambda)=\Re\nu_+^{N+1}(\lambda)$
or $\Re\nu_-^N(\lambda)=\Re\nu_-^{N+1}(\lambda)$.  Note that the
absolute spectrum is not necessarily part of the essential
spectrum. Still, from this definition, the absolute spectrum is
defined by the operator posed on the unbounded domain $\xi\in\R$. The
reader is invited to check that the absolute spectrum of the
advection-diffusion problem~(\ref{e:rcd}) is actually given by the
half line $(-\infty,\mu-\frac{1}{4}]$. In general, the absolute
spectrum is a closed set and consists of a finite collection of
curves.

An important property of the absolute spectrum is that in its
complement, there exist exponential weights
${\underline{\eta}}(\lambda)=(\eta_-(\lambda),\eta_+(\lambda))\in\R^2$ such that
${\cal L}-\lambda$ is Fredholm with index zero when considered as a
closed operator on $L^2_{{\underline{\eta}}(\lambda)}$, the space of functions in
$L^2_{\rm loc}$ with
\[
\|u\|_{L_{{\underline{\eta}}}^2}^2:=\int_{\R^+}|\rme^{\eta_+\xi}u(\xi)|^2
+\int_{\R^-}|\rme^{\eta_-\xi}u(\xi)|^2 <\infty.\] In
Theorem~\ref{t:trunc} below we characterize the limiting behavior of
spectra, as the domain size tends to infinity.  We will consider
periodic boundary conditions, $u(-L)=u(L), \,u_\xi(-L)=u_\xi(L)$, as well
as separated boundary conditions, $(u(\pm L),u_\xi(\pm L))\in E_\pm$
with given, $N$-dimensional subspaces $E_\pm$ of $\C^{2N}$. Periodic
boundary conditions are understood with the restriction $a_+=a_-$,
that is, we are interested in linearizations about pulses.

The {\em point spectrum}, ${\rm spec\,}_{\rm pt}({\cal L})$, refers to
the point spectrum in $L^2(\R)$. For the case of separated boundary
conditions, we distinguish two types of point spectrum. Fix
$\lambda_*\not\in {\rm spec\,}_{\rm abs}({\cal L})$, and the weight
$(\eta_-^*,\eta_+^*)=\underline{\eta}_*=\underline{\eta}_*(\lambda_*)$
such that ${\cal L}$ is Fredholm of index zero for $\lambda$ near
$\lambda_*$ in $L^2_{\underline{\eta}_*}$.  We say $\lambda_*$ belongs
to the {\em extended point 
spectrum} ${\rm spec\,}_{\rm expt}({\cal L})$, if ${\cal L}-\lambda$
is not invertible in $L^2_{\underline{\eta}_*}$.  We say
$\lambda_*$ belongs to the {\em right boundary spectrum}, ${\rm
spec\,}_{{\rm bdy},>}({\cal L})$, if the asymptotic, constant
coefficient operator ${\cal L}_+ - \lambda_*$ possesses nontrivial
kernel when considered on $\xi>0$,
in $L^2_{\eta^*_+}(\R_+)$ with boundary condition $E_+$ in
$\xi=0$. Similarly, the {\em left boundary spectrum} is
defined via the kernel of ${\cal L}_- - \lambda_*$ in
$L^2_{\eta^*_-}(\R_-)$ with boundary condition $E_-$ in
$\xi=0$.  The {\em boundary spectrum} ${\rm spec\,}_{{\rm bdy}}{\cal L}$ is
defined as the union of left and right boundary spectra. We refer
to~\cite{SStrunc} for a more geometric and constructive definition of
boundary spectra, involving winding numbers or, alternatively, Evans function
constructions.

Several assumptions are needed in the proof of
the following theorem. Most of them are technical in nature
and satisfied for ``typical'' systems. We therefore  refer the reader
to~\cite{SStrunc} for precise hypotheses. 

\begin{Theorem}\cite{SStrunc}
\label{t:trunc}
Under typical assumptions on the spectrum on the real line, we find
that the spectrum of ${\cal L}$ on $L^2(-L,L)$ converges, as $L\to
\infty$:
\begin{itemize}
\item under periodic boundary conditions,
\[
{\rm spec\,}{\cal L}_{{\rm per},(-L,L)}\longrightarrow {\rm spec\,}_{\rm ess}({\cal
L}) \cup {\rm spec\,}_{\rm pt}({\cal L}) 
;\]
\item under typical, separated boundary conditions,
\[
{\rm spec\,}{\cal L}_{{\rm sep},(-L,L)}\longrightarrow {\rm spec\,}_{\rm abs}({\cal
L}) \cup {\rm spec\,}_{\rm expt}({\cal
L}) \cup {\rm spec\,}_{\rm bdy}({\cal
L}) 
.\]
\end{itemize}
Convergence is understood on bounded subsets of the complex plane in
the symmetric Hausdorff distance. 

Moreover, multiplicity of the eigenvalues is preserved. The number of
eigenvalues, counted with multiplicity, in any fixed open neighborhood
of a point $\lambda_*\in {\rm spec\,}_{\rm ess}({\cal L})$ (or ${\rm
spec\,}_{\rm abs}({\cal L})$, respectively) converges to infinity as
$L\to\infty$. Multiplicities in neighborhoods of points $\lambda_*\in
{\rm spec\,}_{\rm expt}({\cal L}),\,{\rm spec\,}_{\rm pt}({\cal
L}),{\rm spec\,}_{\rm bdy}({\cal L}) $ stabilize and convergence of
eigenvalues is of exponential rate in $L$, there.
\end{Theorem}
The most important, and at first sight surprising, part of the theorem
is that spectra on large intervals with separated boundary conditions
do not approximate essential, but merely absolute spectra. The latter
are in general different from the essential spectrum. However, a
reflection symmetry $x\to -x$ in the problem can force both spectra to
coincide.

The theorem shows that the limit of spectra on bounded intervals is
entirely determined by ``spectral'' information from the unbounded
domain limit, together with information on the boundary
conditions.

Let us return to the bifurcation problem in Theorem~\ref{t:1}. An
intriguing question is, what happens to the instability when
restricting the domain to a bounded interval in an appropriately
comoving frame. This restriction becomes necessary, for example, when
trying to detect instabilities of fronts numerically. The critical
spectrum in Theorem~\ref{t:1} consists of an isolated zero eigenvalue
and of essential spectrum with non-zero group velocity. With a
positive exponential weight $\exp(\eta\xi)$  with
$\eta>0$, on $\xi\in\R$, we can shift the essential spectrum into the stable complex
half plane. By Theorem~\ref{t:trunc}, the only critical eigenvalue in
a large domain is going to be the zero eigenvalue. Its precise
location depends on the actual choice of boundary conditions. In
particular, there is no Hopf bifurcation in any, arbitrarily large but
finite, domain caused by the essential spectrum. The only possible
instability mechanism would be the creation of unstable spectrum
through the boundary conditions, in ${\rm spec}_{\rm bdy}{\cal L}$.


Linearized stability, $\Re{\rm\,spec}\,{\cal L}_*<0$, in large bounded
domains $(-L,L)$ implies nonlinear stability, for reaction-diffusion
systems. However, our above result on the spectrum for $L\to\infty$
makes extensive use of exponential weights, which might be
incompatible with the nonlinearity. For example, with
$\eta:=\eta_+=\eta_->0$, the nonlinearity $u\mapsto u^2$ is not
defined on $H^1_\eta$, the space of functions $u$ with $u,u'\in
L^2_\eta$. This technical observation is closely related to
limitations of the above theorem. For an illustration, let us return
to
\begin{equation}\label{e:sc}
u_t=u_{xx} + u_x +\mu u +  \kappa u^3,\quad u(-L)=u(L)=0
\end{equation}
with $L$ large. For $0<\mu<1/4$, the absolute spectrum is contained in
the open left half plane and $u(x)\equiv 0$ is asymptotically
stable. Define the basin of attraction $B_L$ as the set of initial values
$u_0\in H^1_{\rm Dir}(-L,L)$ such that the solution of~(\ref{e:sc}) with
initial value $u_0$ 
converges to zero. As a measure for stability, we define the
{\em instability threshold}
\[
\delta^{\rm u}(L)=\inf\{\|u_0\|_{H^1};\; u_0\not\in B_L\}
.\]
The instability threshold measures the minimal amount of a perturbation
needed to ``permanently'' drive the system away from the
equilibrium. The smaller $\delta^{\rm u}(L)$, the more sensitive
the equilibrium $u\equiv 0$ will be to perturbations. 

\begin{Proposition}~\cite{SSconv}\label{p:nonlin}
Consider equation~(\ref{e:sc}) with $0<\mu<1/4$. Then the instability
threshold $\delta^{\rm u}(L)$ satisfies
\begin{itemize}
\item[a)] 
$\delta^{\rm u}(L)\geq \delta_0>0$ if $\kappa<0$, for all $L>0$;
\item[b)]
$\delta^{\rm u}(L)=\rmO(1/L)$ if $\kappa>0$.
\end{itemize}
\end{Proposition}
In particular, for the unstable sign (b) of the nonlinearity
$\kappa>0$, the stability predicted from the spectral analysis, with
uniform exponential decay rate $\mu-1/4$, is valid only in a very
small region of phase space, if the size $L$ of the domain is
large. Even small initial conditions may then lead to blow-up in
finite time.  We do not know if the upper estimate $\rmO(1/L)$ on the
instability threshold is optimal in case (b).



%\setcounter{equation}{0}
\section{Two space dimensions: existence of spiral waves} 
\label{s:3}

The spontaneous appearance of rotating spiral-shaped planar patterns
is a most striking phenomenon, both from the experimental and the
theoretical point of view. Spiral wave patterns have been observed for
example in heart tissue \cite{wieros46}, in Belousov-Zhabotinsky (BZ)
reactive media \cite{winfre72,braeng93,muelle93}, in the
CIMA-reaction~\cite{dekepper}, in fluid convection experiments
\cite{plabod96}, and in surface catalysis \cite{ertl93}. From a
mathematical view point, they pose challenges because they arise in
large --- ideally unbounded --- domains; see Section \ref{s:3.1}.

Pioneering work by Wiener, 1946, has been motivated by waves of
electrical excitation in heart muscle tissue. It is well worth
obtaining the somewhat obscure original reference \cite{wieros46}. See
also the overwhelming material collected by \cite{winfre80,winfre87}, and the
references there. One approach initiated by Wiener aims at a direct
geometric description of the spatio-temporal dynamics of the appearing
sharp wave fronts, which far from a core region take the form of
Archimedean spirals. This approach is sometimes called ``kinematic
theory of spirals'', and has been developed further on a mostly formal
level; see for example \cite{keetys92,keener92,Mikhailov} and the
references there. The underlying idea is closely related to the problem
of ``curve shortening'', which has seen significant recent progress;
see \cite{angene90,angene91,gagham86}.

Spiral waves appear both in excitable and oscillatory media. In the
latter, the interpretation changes from a source of excitation waves
to a defect in the pattern of phase waves emitted by the spiral core.
At the center, or core, the oscillation phase cannot be defined in a
continuous way. Along a simple closed path around the core, the phase
of the oscillations winds around the circle once. Small-amplitude
oscillations are amenable to the powerful tool of modulation
equations such as the complex Ginzburg-Landau equation or
$\lambda$-$\omega$-systems; see \cite{KoHo,Hagan,Kop}.

In the following two sections, we expose two simple approaches to this
problem. We first address excitable media and a kinematic description
of the spiral arms in Section~\ref{s:kin}.

In Section~\ref{s:3.1}, we then characterize spiral wave solution to
reaction-diffusion systems more generally, aiming at a mathematical
definition which incorporates the characteristic features of
experimentally observed spirals. The formulation in this section will
be the basis for the spectral discussion in Section \ref{s:3.3}.

Small-amplitude spiral waves arising near Hopf bifurcations in the
reaction kinetics then provide us with specific examples, in
Section~\ref{s:spex}.   

\subsection{Kinematics and its defects}
\label{s:kin}
To state the problem in its most simplistic and, as we will see, yet
inadequate form, we consider a differentiable planar curve
$s\mapsto(r,\varphi)(s)$ in polar coordinates $(r,\varphi).$ Arc
length is denoted by $s$. We assume that the curve propagates in time,
with scalar normal velocity $c_\perp$ which depends on the signed
local curvature $\kappa$ in a strictly monotone way. (We fix the
normal direction to the left of the tangent vector here.) For later
convenience, we write
% 
\begin{eqnarray}
c_\bot &=& {\texttt v}^{-1} (\kappa)
\label{3.1.1}\end{eqnarray} 
to denote the inverse dependence. Let $\theta\in S^1$ denote the angle
which the unit tangent to our curve forms with the horizontal
axis; see Figure \ref{Fig3.2}. Denoting the derivative of $\theta$ with respect to arc length
$s$ by $\dot{\theta}$, we obtain $\dot{\theta}=\kappa$ and, by
(\ref{3.1.1}),
%
\begin{eqnarray}
\dot{\theta} &=& {\texttt v}(c_\perp).
\label{3.1.2}\end{eqnarray} 
If the curve is to rotate left at constant angular velocity $\omega$
with respect to the center $r=0$ of our polar coordinate system
$(r,\varphi)$, then
% 
\begin{eqnarray}c_{\perp} &=&\omega r\cos (\theta-\varphi). \label{3.1.3}\end{eqnarray}
Combining (\ref{3.1.2}), (\ref{3.1.3}) with the definition of arc
length parameterization we obtain the ODE system
% 
\begin{eqnarray}\begin{array}{lcl} \dot{\theta} &=&  {\texttt v}(\omega r \cos(\theta-\varphi))\\
\dot{\varphi} &=& r^{-1} \sin(\theta-\varphi)\\ \dot{r} &=&
\cos(\theta-\varphi) \end{array} \label{3.1.4}\end{eqnarray} for any
rigidly rotating curve $(r(s),\varphi(s),\theta(s))$ under the geometric
equation (\ref{3.1.1}).

Curve shortening as mentioned above corresponds to the special case
${\texttt v}(c_\bot) =\kappa$ of normal velocity proportional to
$\kappa.$ As for all $ {\texttt v}$ of positive slope, a Sturm
property analogous to Proposition \ref{p-2.1} holds here: the number
of strict crossings of two distinct curves never increases under this
curvature flow. We note here that positive $ {\texttt v}$-slope, $
{\texttt v}'>0$, corresponds to a nonlinear forward parabolic equation
for the curve evolution; see \cite{angene91}. The existence of
rotating wave "Ying-Yang" spiral curves seems to be folklore in the
subject. The resulting spirals, however, are not Archimedean: they
limit from inside onto a circle of finite radius, instead. Negative
slope, in contrast, would correspond to a backwards parabolic equation
which is ill-posed in forward time. Although this may suggest an
incompleteness in the model derivation of the curve-``lengthening''
equation, it does not cause difficulties for our rotating wave ODE
(\ref{3.1.4}). Variants involving a power dependence like ${\texttt
v}^{-1}(\kappa) =\kappa^\beta$ have also been considered: most notably
the affine invariant case $\beta=1/3$ for self-focusing edge
enhancement in image processing. For propagation of
Belousov-Zhabotinsky (BZ) fronts, a good model is generally believed
to be given by
% 
\begin{eqnarray}{\texttt v}^{-1}(\kappa) &=& c_0 +\kappa \label{3.1.4b}\end{eqnarray}
on suitable time scales, see \cite{keetys92,Mikhailov} and the
references there. 

\subsubsection{Curvature flow of Archimedean spirals}
Let us look for {\em Archimedean spirals }in system (\ref{3.1.4}), as
are observed in BZ media. We consider Archimedean spirals to be
characterized by an asymptotically proportional dependence of radius
$r$ on angle $\varphi$, given by a limiting slope
% 
\begin{eqnarray}\sigma_* := \lim_{r\rightarrow \infty} \frac{dr}{d\varphi} &=& \lim_{s\rightarrow
\infty} \frac{r\cos\alpha}{\sin \alpha} \not\in \{0,\pm\infty\}. \label{3.1.5}\end{eqnarray}
Here we have used l'Hospital in (\ref{3.1.4}), with the abbreviation
\begin{eqnarray}\alpha &:=& \theta - \varphi. \label{3.1.6}\end{eqnarray}
Our curve evolution is in fact rotation invariant, as is
$\alpha=\theta-\varphi$ but neither $\theta$ nor $\varphi$
individually. It is therefore convenient to rewrite (\ref{3.1.4}) in
terms of $(r,\alpha)$:
% 
\begin{eqnarray}
\begin{array}{rcl} \dot{\alpha} &=& {\texttt v}(\omega r \cos \alpha)-r^{-1}\sin \alpha\\
\dot{r} &=& \cos \alpha. \end{array}
\label{3.1.7}\end{eqnarray} 
From (\ref{3.1.5}) in the limit $s\rightarrow\infty,$ alias
$r\rightarrow\infty,$ we immediately deduce that $\cos\alpha
\rightarrow 0$ and in fact
% 
\begin{eqnarray}\begin{array}{rcl} \alpha & \rightarrow &\pm\pi/2\\
\sigma := r\cos \alpha & \rightarrow &\pm\sigma_*. \end{array} \label{3.1.8}\end{eqnarray} 
with corresponding signs $\pm$ for either limit. To analyze these limits further, we rewrite
(\ref{3.1.7}) in terms of $r$ and $\rho:=1/r\rightarrow 0$ as follows
% 
\begin{eqnarray}\begin{array}{rcl} \sigma' &=& \pm\sqrt{1-\rho^2\sigma^2}\; {\texttt v} (\omega\sigma)+\rho\\
\rho' &=& -\rho^4\sigma \end{array} \label{3.1.9}\end{eqnarray} Here
we have rescaled ``time'' $s$ such that $\rho \rmd/\rmd s =\,'.$ The vector
field (\ref{3.1.9}) is regular along the invariant line $\rho=0,$
where
% 
\begin{eqnarray}\sigma' &=& \pm {\texttt v}(\omega \sigma). \label{3.1.10}\end{eqnarray}
The limit slope (\ref{3.1.5}) therefore requires the equilibrium
condition
% 
\begin{eqnarray}0 &=& {\texttt v}(\omega \sigma_*) \label{3.1.11}\end{eqnarray}
to hold. This condition amounts to a dispersion relation between the
rotation frequency $\omega$ and the wave length $2\pi\sigma_*$ of the
asymptotically uncurved, planar fronts of our Archimedean spirals. For
example $\sigma_*=-c_0/\omega$ in the BZ case mentioned above.

A center manifold analysis at the Archimedean spiral equilibrium
$\rho=0,\; \sigma=\sigma_*$ of (\ref{3.1.9}) yields additional
information on those Archimedean spiral trajectories.  Consider the
case $\alpha\rightarrow +\pi/2,\; \sigma_*>0$ of left outward winding
spirals, first. Then $\sigma=\sigma_*$ is hyperbolically stable,
within the line $\rho=0,$ if
% 
\begin{eqnarray}\omega {\texttt v}'(\omega\sigma_*) <0. \label{3.1.12}\end{eqnarray}
Since $\rho\searrow 0$ inside the center manifold, any
initial condition of (\ref{3.1.9}) with $\rho >0$ and $(\rho,\sigma)$ near
$(0,\sigma_*)$ then converges to this Archimedean spiral
equilibrium. Most notably
this includes the BZ example, where ${\texttt v}'>0,\;\sigma^* >0,$ and
$\omega<0.$ The opposite case $\omega {\texttt v}'(\omega \sigma^*) >0$ arises
for example when curves shorten due to ${\texttt v}'>0,\;
c_0<0<\omega.$ 
Then $\sigma=\sigma_*$ is hyperbolically unstable, within
the line $\rho=0$, and the asymptotically Archimedean spiral is
uniquely defined. In either
case, the center manifold --- be it unique or not --- provides
convergence to the Archimedean spiral equilibrium along the same
asymptote
% 
\begin{eqnarray}
\sigma=\sigma (r) \sim \sigma_* -(\omega {\texttt v}'(\omega\sigma_*))^{-1}
r^{-1} + O (r^{-2}). 
\label{3.1.13}\end{eqnarray} 
The $r^{-1}$-term measures the deviation from a precise Archimedean
spiral. Similarly, polar coordinate asymptotics for $r=r(\varphi)$ can
be derived.

\subsubsection{The front-back matching problem}
The two sign-cases in (\ref{3.1.9}) are related by a simple
transformation. Indeed $(\rho,\sigma,s)\mapsto (-\rho,\sigma, s)$
reverses the sign $\pm$ as well as the sign of $'$. This means that
the unique equilibrium $\sigma=\sigma_*$ on the line $\rho =0$
possesses the same stability properties under the $s$-flow, in either
case. In particular it is impossible to find a rotating wave solution
as in Figure \ref{Fig3.2}(a), which would be biasymptotic to an
Archimedean spiral for both $s\rightarrow +\infty$ and $s\rightarrow
-\infty$.

This impossibility indicates a first defect of our naive, inadequate
approach based on motion by curvature (\ref{3.1.1}) alone. Two
appealing remedies have been proposed. The first derives curvature
motion for fronts $(s\rightarrow +\infty)$ and backs $(s\rightarrow
-\infty)$ separately, by singular perturbation arguments, and then
attempts a formal matching in the ``core'' region where the arc length
parameter $|s|$ is small. See for example \cite{keener92}.

%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,3.5)%
\centerline{\includegraphics[scale=0.85]{Fig/Fig15.eps}}%used to be Fig12.eps
\end{picture}%
\caption{Biasymptotic Archimedean spiral (a), and spiral with spiral tip (b). } 
\label{Fig3.2}
\end{figure}


More radically the second remedy terminates the spiral at $s=0$ and
proposes additional phenomenological equations of motion at this
``tip'' point, see for example \cite{Mikhailov}, and Figure
\ref{Fig3.2}(b). Neither approach has been fully justified on a
satisfactory mathematical basis, so far.
For scalar equations, $u\in \mathr$, the singular perturbation limit
$\varepsilon\searrow 0$ for reaction-diffusion equations
% 
\begin{eqnarray}\varepsilon u_t &=& \varepsilon^2 \Delta u + f(u) \label{3.1.14}\end{eqnarray}
has been analyzed successfully for bounded domains $x\in\Omega \subset
\mathr^N,$ and (mostly) cubic nonlinearities
% 
\begin{eqnarray}f(u) &=& (u-a)(1-u^2). \label{3.1.15}\end{eqnarray}
Concerning planar wave fronts, alias $N=1,$ between the stable
equilibria $u\equiv \pm 1$ we have already mentioned the travelling
waves $u=u((x-c_0 t)/ \varepsilon)$ of wave speed $c_0=c_0(a).$ Note
that $c_0(-a)=-c_0(a),$ by the symmetry $u\mapsto -u.$ Based on
earlier formal expansions by Fife, an asymptotics
% 
\begin{eqnarray}c_\perp &=& c_0(a)+\varepsilon \kappa + \cdots \label{3.1.16}\end{eqnarray}
has been proved to hold for $N=\dim x=2$, over time spans $0\leq t\leq
T/\varepsilon,$ for the interface curves of solutions
$u=u^\varepsilon $ of (\ref{3.1.14}). Mean curvature flow for
$(N-1)$-dimensional interfaces has also been justified, for $N\geq 3.$
See \cite{fife88,barlesetal93} and the references there.

Expansion (\ref{3.1.16}) fits well with the BZ-Ansatz (\ref{3.1.4b})
for $c_\perp={\texttt v}^{-1}(\kappa).$ Choosing $a=a(\varepsilon)=\varepsilon
a_1$ we can in fact scale $\varepsilon$ out of (\ref{3.1.16}) by
rescaling time, because oddness of $c_0(a)$ then implies
$c_0(a)=\varepsilon c_0'(0)a_1+\cdots\;.$ Then (\ref{3.1.16}) takes
precisely the form (\ref{3.1.4b}) discussed above.

However, rigorous analysis of (\ref{3.1.14}) is heavily based on
variational methods and hence fails, as it stands, in two important
cases: in unbounded domains $\Omega$ because the Lyapunov functional
${\cal V}={\cal V}_\varepsilon $ of (\ref{2.1.3}) may become
unbounded, and for systems $u\in \mathr^2$ because ${\cal V}$ may not
exist. 

The front-back difficulty, which causes the inexistence of
biasymptotic Archimedean spirals as in Figure \ref{Fig3.2}(a), already
arises in one space dimension $N=1$ and is due to the planar front
wave speeds $c_0(a)$ in (\ref{3.1.4b}), (\ref{3.1.16}). Indeed,
adjacent curve segments in Figure \ref{Fig3.2}(a) correspond to limits
$s\rightarrow\pm\infty,$ alternatingly, where curvature $\kappa$ is
negligible and
% 
\begin{eqnarray}c_\perp &\approx & c_0(a) \label{3.1.17}\end{eqnarray}
approaches the planar wave front limit. But the normal directions
point in opposite directions, for adjacent segments. Alternatively,
with respect to a fixed normal direction or as is appropriate in one
space dimension $N=1,$ we can invoke $c_0(-a)=-c_0(a)$ and observe
that adjacent fronts and backs move in opposite directions, unless
$c_0(a)=0.$ That latter possibility is excluded, both by experimental
observation and by our constitutive assumption $\sigma_*\neq 0$ for
Archimedean spirals, in (\ref{3.1.5}).

As an alternative to this dilemma we might feel tempted to consider
travelling pulses, for $N=1$, instead of fronts and backs. Rescaling
$u=u((x-c_0t)/\varepsilon)$ by $\varepsilon \searrow 0,$ the pulse
locations would contract to curves, for $N=2,$ which could move
according to curvature. For scalar equations, $u\in\mathr$ however,
this is prevented by the (damped) Hamiltonian structure of the
travelling wave equation, which forces vanishing wave speed $c_0=0$ for
any homoclinic pulse. Placing a single pulse on any large circle $r=
const.,$ moreover, the curve of pulse locations would have to
terminate at a ``tip'' as in Figure \ref{Fig3.2}(b).

Clearly this Hamiltonian difficulty disappears for non-variational systems
$u\in\mathr^2.$ We conclude with a standard example for excitable media systems: the
FitzHugh-Nagumo system
% 
\begin{eqnarray}\label{3.1.18} 
u^1_t &=& \Delta u^1+\varepsilon^{-1}u^1(1-u^1)(u^1-(u^2+b)/a)\nonumber\\
u^2_t &=& d\Delta u^2 + u^1 - u^2. 
\end{eqnarray} 
For given $u^2=u^2(t,x),$ the singularly perturbed $u^1$-equation is
cubic as discussed above, and $u^2$ enters as a parameter. Adjusting
the constants $a,b$ appropriately, homoclinic pulses arise which
travel at nonzero speeds. This case is usually termed {\em excitable
medium}; see Figure \ref{Fig3.3}(a).
%
\begin{figure}%[h]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,4.)% 
\centerline{\includegraphics[scale=1.0]{Fig/Fig13.eps}}
\end{picture}%
\caption{Excitable (a) and oscillatory (b) media; homoclinic pulses
and periodic travelling waves for $\varepsilon \searrow 0.$ Isoclines
in absence of diffusion. See also (\ref{3.1.18}) for the excitable
case (a). } \label{Fig3.3}
\end{figure}


The $u^2$-values are nearly constant through each $u^1$-jump, in the
singular limit $\varepsilon \searrow 0.$ The diffusive $u^2$-equation,
however, constitutes a highly nonlocal and nonlinear coupling of the
appropriate local wave speeds $c_0=c_0(u^2)$ in the curvature motion
(\ref{3.1.4b}) or (\ref{3.1.16}). It seems tempting to derive an
equation for the tip motion, in this case, involving local information
on the ``pulse'' curve at the tip itself. See \cite{Mikhailov} for a
phenomenological attempt.


Spatially periodic stable travelling waves can likewise be realized,
for appropriate $a,b$ and small $\varepsilon,$ if we modify the
equation $u^2_t$ by subtracting a suitable positive constant; see also
Figure \ref{Fig3.3}(b). This corresponds to an {\em oscillatory medium.}

In the following section we will take an entirely different approach,
based on reaction-diffusion systems in the plane $x\in
\Omega=\mathr^2,$ but neither restricted to, nor complicated by,
singular perturbations. In particular, in Section \ref{s:spex}, we
address the issue of primary bifurcation of spiral waves from
spatially homogeneous equilibria in the presence of continuous
spectrum. In Chapter~\ref{s:4}, we will further pursue this approach
to include secondary bifurcations from Archimedean rotating waves.

\subsection{Archimedean spiral waves in radial dynamics}
\label{s:3.1}
In Section~\ref{s:3.1.1}, we collect some basic characteristics of
spiral waves in reaction-diffusion systems: rigid rotation and
Archimedean shape. Section~\ref{s:2.1a} clarifies, in which sense
spiral waves ``emit'' wave trains. We summarize some
principal features of a spiral wave, in Section~\ref{s:de}.

\subsubsection{Rigid rotation and asymptotic wavetrains} 
\label{s:3.1.1} 
Proving existence of spiral waves in reaction-diffusion systems has
been a long standing mathematical problem. In spite of their
prevalence and robustness in experiments as well as in numerical
simulations, proofs of existence for spiral waves are rare.  Given the
empirical evidence for existence, one might suspect that the existence
problem is of purely mathematical interest.  We believe, however, that
a proper understanding of the existence problem also leads to a
systematic approach to many observable stability and instability
phenomena involving spiral wave dynamics.

Consider a reaction-diffusion system in $x\in\R^2$, the unbounded
plane,
\begin{equation}
\label{e:rdsys}
u_t=D\Delta_x u + f(u),\quad x\in\R^2
.\end{equation}
We aim at a precise definition of what we understand by an Archimedean
spiral wave solution to (\ref{e:rdsys}).

First of all, the simplest  spiral waves observed in chemical
experiments are rigidly rotating. Rigid rotation is easily described in
polar coordinates $(r,\varphi)$, as we saw in Section~\ref{s:kin}. The
central point of our discussion of Archimedean spirals in Chapters
\ref{s:3} and \ref{s:4}, on top of that mere convenience, will be a
particular view on the radial coordinate $r$: we will view radius $r$,
rather than time $t$, as a new, spatial ``time'' variable --- much in
the same way as we have viewed the spatial variable $x$ as ``time'' in
our discussion of travelling waves in Chapter~\ref{s:1u}.  We refer to
this view point, which was first used in~\cite{Sspiralex} for the
spiral wave problem and further exploited in~\cite{Shabil}, as {\em
radial dynamics}.

In polar coordinates $(r,\varphi)$, rigid rotation allows us to write
the spiral wave solution $q_*$ in the form
\begin{equation}\label{e-sppc}
u(t,r,\varphi)=q_*(r,\varphi+\omega_{*} t).
\end{equation}
Here, $\omega_{0}$ is the temporal frequency of rotation of the
spiral. Inserting this Ansatz into the reaction-diffusion system, we
find 
\begin{equation}\label{e:ellsp}
D(u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2}u_{\psi\psi}) - \omega_{*}
u_\psi + f(u)=0,\end{equation} a partial differential equation of
elliptic type for $u=q_*(r,\psi)$, with
$\psi=\varphi+\omega_*t$. Note, however, that the operator
$\omega_*\partial_\psi$ is not relatively bounded with respect to the
Laplacian. Indeed, the derivative $\partial_\psi$ cannot be bounded by
$ r^{-2}\partial_{\psi\psi}$. As a consequence, the smooth function
$\sin(x_1)$, for example, belongs to the domain of the Laplacian on
spaces of bounded, uniformly continuous functions, but the angular
derivative $x_2\partial_{x_1} \sin(x_1)$ is unbounded.

An additional characteristic of many observed spiral waves is their
Archimedean shape. We require the profile to be Archimedean far away
from the center of rotation,
\begin{equation}\label{e:asyarch}
|q_*(r,\psi)-q_{\infty}( r-(\psi/k_{\infty}))|\longrightarrow
0\quad\mbox{for } r\to\infty,
\end{equation} 
uniformly in $\psi$, for some function $q_{\infty}(\xi)$. Note that
$q_\infty$ is constant along Archimedean spirals $\psi=k_\infty r
+\psi_0$. Convergence (\ref{e:asyarch}) is the ``boundary condition'' that
we will impose at $r=\infty$ for our spiral wave solution to the
elliptic system~(\ref{e:ellsp}). The asymptotic profile of the spiral
wave, $q_{\infty}$, has to satisfy certain compatibility
conditions. First, $q_{\infty}$ is $2\pi/k_{\infty}$-periodic in its
argument by continuity. Second, if we formally let $r\to\infty$
in~(\ref{e:ellsp}), we find the condition
\begin{equation}\label{e:bdy}
 D q''_{\infty} +\frac{\omega_*}{ k_{\infty}} q'_{\infty} +
f(q_{\infty})=0.
\end{equation} 
This is nothing else but the travelling-wave equation for
one-dimensional periodic wavetrain solutions $q(x-(\omega_*/k_{\rm
wt}) t)$ of
\[
u_t = Du_{xx}+ f(u),
\]
with speed $\omega_*/k_\infty$ and spatial period $2\pi/k_\infty$.
We therefore call $q_\infty$ the {\em asymptotic wavetrain} of the spiral
wave solution. The constant $k_{\infty}$ appearing in (\ref{e:asyarch})
is the wavenumber of the asymptotic  wavetrain. The speed of the
asymptotic wavetrain is computed from the standard relation 
\begin{equation}\label{e:cok}
c_{\infty}=\omega_*/k_{\infty}.
\end{equation} 
For fixed, arbitrary, speed $c$, periodic solutions to the travelling-wave
equation~(\ref{e:bdy}) are typically isolated. The spatial period
$2\pi/k$ depends nontrivially on $c$. Using
(\ref{e:cok}) with $k_\infty=k$, $c_\infty=c$, we can turn this
dependence in a relation between $\omega$ and $k$. We call
this relation the {\em nonlinear dispersion relation} of the asymptotic
wavetrain
\begin{equation}\label{e:nldr}
d_{\rm nl}(\omega,k)=0.
\end{equation}
The spiral ``selects'' a particular solution $\omega=\omega_*$,
$k=k_\infty$ of (\ref{e:nldr}). 

\subsubsection{Linear and nonlinear group velocities}
\label{s:2.1a}
Most spiral waves that have been observed in chemical experiments share
an additional property: wavetrains are ``emitted'' by the center.
Spirals with this property are sometimes referred to as active
spirals. Mathematically, the emission of wavetrains is characterized
by a positive group velocity $c_{\rm g}>0$ of the asymptotic
wavetrains in the radial direction. The concept of group velocity we
are referring to, here, is a slight generalization of the definition
in equation~(\ref{e:cg}).  There are actually two equivalent
definitions of the group velocity for wavetrains. The simplest
definition would refer to the nonlinear dispersion
relation~(\ref{e:nldr}). Assuming that we can solve for
$\omega_{\infty}$ as a function of $k_{\infty}$, we define the
{\em nonlinear group velocity} through
\begin{equation}\label{e:cgnl}
c_{\rm g,nl}=\frac{\rmd \omega_{\infty}}{\rmd k_{\infty}}
,\end{equation}
evaluated at the wavenumber of the wavetrain, $k_{\infty}$.

We say that a spiral waves {\em emits} the asymptotic wavetrain, if
the nonlinear group velocity of the asymptotic wavetrain is positive.

Another definition of the group velocity would refer to the linearization about the
wavetrain $q_{\infty}$ in a comoving frame of speed
$\omega_*/k_{\infty}$,
\begin{equation}\label{e:5.27}
w_t = Dw_{\xi\xi} + \frac{\omega_*}{k_{\infty}}w_\xi + f'(q_\infty(\xi))w.
\end{equation}
At the end of
Section~\ref{s:1.2}, (\ref{e-fl})--(\ref{e:drper}), we explained how
to solve  this linear parabolic equation with spatially periodic
coefficients by means of a generalized Fourier transformation. Indeed,
the general solution can be decomposed into Bloch waves
$w(t,\xi;k)$,
\begin{equation}\label{e:5.28}
w(t,\xi;k)=\rme^{\lambda(\rmi k)t - \rmi k\xi} w_0(\xi;k),
\end{equation}
where $w_0$ solves
\begin{equation}\label{e:cglin0}
(\lambda(\rmi k)+c_{\infty}\rmi k)w_0 = D(\frac{\rmd}{\rmd\xi}-\rmi
k)^2 w_0 + c_\infty \frac{\rmd}{\rmd\xi}w_0 + f'(q_\infty(\xi))w_0.
\end{equation}
Here, $w_0(\xi; k)=w_0(\xi+(2\pi/k_{\infty}); k)$ is spatially
periodic with the period of the wavetrain $q_\infty$ and the Bloch
wavenumber $k$ introduces a second spatial period. The eigenvalue
$\lambda$ and the Bloch wavenumber $ k$ are related by a dispersion
relation $d_{\rm lin}(\lambda,\rmi k)=0$, see
(\ref{e:drper}). Assuming that we can solve this equation for
$\lambda=\lambda(k)$, with $\lambda(0)=0$, the linear group
velocity is computed in analogy to (\ref{e:cg})
\begin{equation}\label{e:cglin}
c_{\rm g,lin}= c_{\infty} + \left.\Im\frac{\rmd\lambda}{\rmd
 k}\right|_{ k=0} 
.\end{equation}
The summand $c_{\infty}$ accounts for the fact that we compute the
dispersion relation in a comoving frame, whereas we are interested in
the group velocity in the steady frame. Note that we can define a
linear group velocity for different values of $ k$, and actually
on different solution branches $\lambda^\ell(\rmi  k)$ of the dispersion
relation $d_{\rm lin}$, again in analogy to (\ref{e:cg}).  

\begin{Lemma}\label{l:l=n}
Linear and nonlinear group velocities of periodic wavetrains, as
defined in (\ref{e:cgnl}) and (\ref{e:cglin}), coincide. In
particular, whenever one of the two is well-defined, both are
well-defined and 
\[
c_{\rm g,nl}=c_{\rm g,lin}=:c_{\rm g}.
\]
\end{Lemma}
The proof of the lemma reduces the computation of both group
velocities to a boundary-value problem, which can then readily be
solved by Lyapunov-Schmidt reduction.

\subsubsection{Characterizing Archimedean spirals}
\label{s:de}
In Sections~\ref{s:3.1.1} and \ref{s:2.1a} we have collected three
properties of certain chemical spiral waves: rigid rotation,
asymptotic Archimedean shape, and positive group velocity of the
asymptotic wavetrains. Heading for an open class of ``robust''
solutions that resemble Archimedean spirals, we incorporate the
apparent robustness of spiral waves with respect to changes in
parameter values into our characterization. So far, we have set up a
boundary-value problem for spiral wave solutions. Preferably,  robustness
proofs for spiral would rely on an implicit function theorem
in a suitable function space. Unfortunately the properties of spiral
waves, as collected above, prevent us from an application of the
implicit function theorem, since the linearization of (\ref{e:ellsp})
lacks the Fredholm property.

More specifically, consider an Archimedean spiral wave solution
$q_*(r,\psi)$ to (\ref{e:ellsp}) on $x\in\R^2$. Linearizing
(\ref{e:rdsys}) at  $q_*$ in a corotating frame yields
\begin{equation}\label{e:splin}
w_t=D\Delta_{r,\psi} w -\omega_* \partial_\psi w + f'(q_*(r,\psi))w=:{\cal L}_*w.
\end{equation} 
The elliptic operator ${\cal L}_*$ is not Fredholm on
$L^p$-spaces or spaces of continuous functions; see~\cite[Lemma
6.4]{SSW2}, for example.  We therefore cannot conclude persistence of
a spiral wave solution under slight variation of diffusion
coefficients or reaction kinetics, simply by an implicit function
theorem.

In the remainder of this section, we aim at a definition of a spiral
wave that will allow us to conclude robustness. The main idea is to
invoke Lemma~\ref{l:cgweight}, relating Fredholm properties in
exponentially weighted spaces to group velocities. Analogously to the
weighted $L^2$-spaces in (\ref{e:expweight}) on the real line
$x\in\R$, we introduce the exponentially weighted spaces
\begin{equation}\label{e:spexp}
L^2_\eta(\R^2):=\{u\in L^2_{\rm
loc}(\R^2);\;\|u\|^2_{L^2_\eta}:=\int_{\R^2} |\rme^{\eta |x|}u(x)|^2
<\infty\},\end{equation} for weights $\eta\in\R$ in the radial
direction $r=|x|$.
\begin{Definition}
\label{d:sw}
An {\em Archimedean spiral} is a bounded rotating-wave solution
$q_*(r,\varphi-\omega_*t)$ of the reaction-diffusion
equation~(\ref{e:rdsys}), with some nonzero rotation frequency
$\omega_*$, which converges to plane wavetrains in the farfield
\[
|q_*(r,\psi)-q_{\infty}(r- (\psi/k_{\infty}))|\longrightarrow 0
,\quad \mbox{ for } r\to\infty,
\]
uniformly in $\psi\in[0,2\pi]$. We call an Archimedean spiral {\em
transverse}, if the following conditions hold:
\begin{enumerate}
\item there exists $\eta_0<0$ such that the linearization ${\cal
L}_{*}$, defined in~(\ref{e:splin}), is Fredholm with index 0 in
$L^2_\eta$, for all $\eta_0<\eta<0$, and the dimension of the
generalized kernel is one;
\item there is $\eta_1>0$ such that the linearization ${\cal L}_{*}$
is Fredholm with index -1 for all $\eta$ with $0<\eta<\eta_1$.
\end{enumerate}
\end{Definition}
The definition encodes positive group velocity of the wavetrain in the
negative Fredholm index of the linearization; see
Proposition~\ref{p:ivc}, Lemma~\ref{l:cgweight}, and the discussion of
the linearization in Section~\ref{s:3.3}, below. It also reflects
simplicity of the trivial zero eigenvalue $\lambda=0$. For fronts and
pulses, we have taken advantage of this property in order to conclude
transverse crossing of stable and unstable manifolds in the
travelling-wave equation, see
Section~\ref{s:1.3}. It is for this geometric interpretation that we
refer to Archimedean spirals satisfying Definition~\ref{d:sw}(i),(ii)
as transverse Archimedean spirals.

\begin{Theorem}\cite{SSspiralstability}\label{t:sprob}
Assume the reaction-diffusion system~(\ref{e:rdsys}) with analytic
reaction kinetics $f=f(u)$ possesses a transverse Archimedean spiral
wave solution. Then, for any reaction kinetics $\tilde{f}$,
$C^1$-close to $f$, and any diffusion matrix $\tilde{D}$ close to
$D$, the perturbed reaction-diffusion system 
\[
u_t=\tilde{D}\Delta u + \tilde{f}(u)
\]
also possesses a transverse Archimedean spiral wave.
\end{Theorem}
For the proof of the theorem, we rewrite~(\ref{e:ellsp}) in radial
dynamics using $r=|x|$ as ``time''; see~\cite{Sspiralex}. We then
interpret a spiral wave as a heteroclinic front solution connecting
the center $r=0$ with the wavetrains in the farfield
$r=\infty$. The radial dynamics formulation of~(\ref{e:ellsp}) reads
\begin{eqnarray}\label{e:exsp}
U_r&=&V\\ V_r&=&-\frac{1}{r} V - \frac{1}{r^2}\partial_{\psi\psi} U -
D^{-1}(-\omega_*\partial_{\psi}U + f(U)). \nonumber
\end{eqnarray}
Note how spiral waves $q_*(r,\psi)$ actually turn into
heteroclinic orbits $(U,V)(r,\cdot)$, just like fronts in one
space-dimension, Section~\ref{s:1}, or the modulated fronts in
Section~\ref{s:1.3}. We view equation~(\ref{e:exsp}) as a
non-autonomous differential equation on a Hilbert space
$(U,V)(r,\cdot)\in H^1(S^1)\times L^2(S^1)$.  Just like the
travelling-wave ODE for modulated fronts~(\ref{e:spatdyn}), the
initial value problem for the elliptic system~(\ref{e:exsp}) is
ill-posed. Stable and unstable manifolds exist, but turn out to be
infinite-dimensional. An additional complication arises since the term
involving the highest derivative $\frac{1}{r^2}\partial_{\psi\psi}U$
is non-autonomous, singular at the origin $r=0$, and degenerate in the
farfield $r=\infty$. 

The limit $r\to 0$ of the core region is easily understood in the
scaling $r=\rme^\tau$, where we find
\begin{eqnarray}\label{e:exsp0}
U_\tau&=&W\\
W_\tau&=&\partial_{\psi\psi}U -\rme^{2\tau}D^{-1}(-\omega_*\partial_\psi U +
f(U)).\nonumber 
\end{eqnarray}
In the farfield limit $r\to\infty$, we recover the equation for modulated
travelling waves (\ref{e:spatdyn}),
\begin{eqnarray}\label{e:exspinf}
U_r&=&V\\
V_r&=& -D^{-1}(-\omega_*\partial_\psi U + f(U)).\nonumber
\end{eqnarray}
Spiral waves are solutions which converge  to constant functions
$U=U_0,V=0$ in
the center of rotation $r=0$. For
$r\to\infty$, they approach the asymptotic wavetrain $U=q_\infty(r-(\psi/k_{\infty})),V=u'_{\infty}(r-(\psi/k_{\infty}))$. In this sense,
spiral waves are heteroclinic orbits in the non-autonomous
differential equation~(\ref{e:exsp}), connecting an equilibrium at
$\tau=\log r =-\infty$ to a periodic orbit at $r=+\infty$. 

Transversality of Archimedean spirals, in the sense of
Definition~\ref{d:sw}, implies that the associated heteroclinic orbit
in spatial dynamics consists of a {\em transverse crossing} of the
stable manifold of the asymptotic periodic orbit, at $r=\infty$, and
the unstable manifold at $r=0$. Transverse crossing, here, again
refers to a transverse intersection of stable and unstable manifolds
in the extended phase space, augmented by the spiral frequency
parameter $\omega_*$; see also Section~\ref{s:sdei}. We return to
this geometric picture for the linearized equation in more detail in
Section~\ref{s:3.3}, below.

The proof of Theorem~\ref{t:sprob} is therefore based on smoothness of
center-stable and center-unstable manifolds in the radial dynamics
formulation, and a robustness argument for transverse
intersections. Analyticity of $f$ is used for uniqueness of solutions
to the ill-posed initial-value problem in the radial dynamics
formulation.  

\subsection{Bifurcation to spiral waves}
\label{s:spex}
Theorem~\ref{t:sprob} shows that transverse Archimedean spiral waves
are robust. We now show that transverse Archimedean spiral waves
actually exist. We construct these spirals through bifurcation from a
spatially homogeneous equilibrium in the unbounded plane $x\in\R^2$, in
the presence of critical continuous spectrum.

We consider reaction-diffusion equations with the pure
reaction-kinetics undergoing a Hopf bifurcation. We assume that
$f(u;\mu)$ depends on a parameter $\mu$ and $f(0;\mu)=0$ for $\mu$
close to zero. The linearization $\partial_u f(0;0)$ is assumed to
possess a pair of purely imaginary eigenvalues $\pm\rmi\omega_{\rm H}$, with
eigenvectors $u_*,\bar{u}_*\in\C^N$. We assume that all other
eigenvalues are contained in the left complex half plane.  Performing
a center-manifold reduction and a subsequent normal-form transformation, the Hopf
bifurcation in the pure kinetics $u_t=f(u;\mu)$ can be reduced to a
two-dimensional ordinary differential equation of the form
\begin{equation}\label{e:hopfnf}
\dot z = \lambda(\mu) z + \beta z|z|^2 + \rmO(|z|)^5,
\end{equation}
with eigenvalue $\lambda(0)=\rmi\omega_{\rm H}$ and complex cubic
normal form coefficient $\beta\in\C$. We assume that $\lambda'(0)>0$,
crossing of eigenvalues, and $\Re\beta<0$. The linearization of the
{\em spatially dependent} reaction-diffusion system in $u=0$,
\[
u_t = D\Delta u + \partial_u f(0;0)u,
\]
can be analyzed after Fourier transform from the dispersion relation
\begin{equation}
\label{e:disphopf}
d(\lambda,\rmi k)={\rm det}\,(-Dk^2+\partial_u f(0;0)-\lambda)=0,\quad k\in\R. 
\end{equation}
For wavenumbers $k\in\R$ near zero, the eigenvalue
$\lambda=\rmi\omega_{\rm H}$ continues to a critical spectral curve
$\lambda(\rmi k;\mu)$, with $\lambda(0;0)=\rmi\omega_{\rm H}$ and
expansion
\begin{equation}
\label{e:hdisp}
\lambda(\rmi k;0)=\rmi\omega_{\rm H}+\alpha k^2+\rmO(k^4).
\end{equation} 

\begin{Theorem}\cite{Sspiralex}\label{t:spex}
There exists a nonempty open set ${\cal U}\subset C^3(\R^N)$ of
nonlinearities $f=f(u)$, such that for all $f\in {\cal U}$ the
reaction-diffusion system~(\ref{e:rdsys}) possesses a transverse
Archimedean spiral wave, in the sense of Definition~\ref{d:sw}. 

More specifically, assume that the reaction-diffusion system undergoes a Hopf
bifurcation such that 
\begin{enumerate}
\item $\lambda(0;0)=\rmi\omega$ is a simple zero of (\ref{e:disphopf})
and the
only purely imaginary solution to (\ref{e:disphopf}), for any $k\in\R$;  
\item $\partial_\mu\Re\lambda(0;0)>0$;
\item for $\alpha$ and $\beta$, as defined in (\ref{e:hopfnf}) and
(\ref{e:hdisp}), we have $\Re\beta<0,\,\Re\beta<0$ and $|{\rm
arg\,}(\beta/\alpha)|<\delta$ sufficiently small.
\end{enumerate}
Then, for sufficiently small $\mu>0$, there exists a transverse
Archimedean spiral wave solution of~(\ref{e:rdsys}).
\end{Theorem}

One approach to Theorem~\ref{t:spex} would be to approximate the
reaction-diffusion system by a complex Ginzburg-Landau equation
\begin{equation}\label{e:cgl}
A_{\tilde t}=\alpha\Delta_{\tilde{x}} A + A + \beta A|A|^2,\quad
\tilde{x}\in\R^2,\ A,\alpha,\beta\in\C,
\end{equation}
and then look for rigidly rotating spiral wave solutions. For
persistence of spiral waves however, we might still have to rely on a
spatial dynamics formulation as in Theorem~\ref{t:sprob}.

The proof given in \cite{Sspiralex} splits into three parts. The first
part contains a reduction of the radial dynamics~(\ref{e:spatdyn}) to a
finite-dimensional center manifold, which contains all solutions
of~(\ref{e:exsp}) close to the equilibrium $(U,V)\equiv
0$~\cite[Theorem 1, Proposition 5]{Sspiralex}. On the center manifold,
we find the bifurcation
equation
\begin{equation}\label{e:redcgl}
A_{rr}=-\frac{1}{r}A_r - \frac{1}{r^2}A + A - \rmi\tilde{\omega} A
+(\beta/\alpha) A|A|^2,
\end{equation}
for $A(r)\in\C$, to leading order in an appropriate scaling. The
parameter $\tilde{\omega}$ depends on the frequency of rotation
$\omega_*$.  In the second part, this reduced equation is shown to
possess a transverse heteroclinic orbit, asymptotic to plane wave
solutions with positive group velocity~\cite{Hagan,Kop,Sspiralex}. The
proof is concluded with a persistence result under higher-order perturbations,
similar to Theorem~\ref{t:sprob}.

The results in~\cite{Sspiralsub,Sspiralex} also show that spatially localized
rotating wave solutions $u(r,\varphi-\omega t)$ with $u(r,\cdot)\to 0$
for $r\to\infty$ bifurcate if the Hopf bifurcation is
subcritical. To our knowledge, these patterns have not been observed in
experiments, yet. 


\section{Two space dimensions: bifurcations from spiral waves} 
\label{s:4} 

In this chapter we discuss several instability mechanisms for the
spiral rotating wave solutions which are constructed in the previous
chapter. We start with a biased review of experimentally observed
instabilities, in Section~\ref{s:3.4}. We then consider instabilities
of Hopf type, caused by point spectrum. Such instabilities lead to
meandering motions of the spiral tip and to periodic shape
fluctuations.  Equivariant bifurcations under the noncompact symmetry
group $SE(m)$ are an appropriate framework here, as we will discuss in
Section~\ref{s:meander}. 

Beyond the two-frequency meandering motion, more complicated motions of the spatial tip have been observed \cite{muelle93}. Attempts to explain these motions strongly rely on a finite-dimensional description of the instability, coupled to the motion on the non-compact group $SE(2)$ \cite{fietur98,Gol,asmeni01}. The motion of the tip, often referred to as hypermeander (although already meander originally refers to an irregular winding of river beds) can under certain assumptions be shown to trace Brownian motion paths; see Section 5.2.7.


In Section~\ref{s:3.3}, we analyze the
linearization about Archimedean spiral waves in a more systematic
way. In particular, we classify continuous spectra and the
super-spiral shapes of eigenfunctions. The results in this section are
then compared with experiments and numerical simulations in
Section~\ref{s:4.n}: the spatio-temporal patterns arising in spiral
breakup and meander instabilities are related to spectral properties
of the primary Archimedean spiral.

The discussion of continuous spectra in Sections \ref{s:3.3} and
\ref{s:4.n} is largely independent of Section~\ref{s:meander} and
closely follows the exposition on radial dynamics in
Sections~\ref{s:3.1}, \ref{s:spex} and on the one-dimensional case,
Chapter~\ref{s:1u}.


\subsection{Phenomenology of spiral instabilities}
\label{s:3.4}

Beyond rigidly rotating spiral waves, destabilization of individual
spiral waves and subsequent transition to still richer spatio-temporal
patterns has been observed in chemical experiments.  In this section,
we review some experimentally and numerically observed instabilities.

Meandering of the spiral tip was observed
in~\cite{Win,SkSw,ertl93,braeng93,muelle93}, among others. Temporally,
the spiral motion becomes a two-frequency motion. For example, in a
two-species reaction-diffusion system $N=2$, we may trace the
evolution of a point $x=z_*(t)$ in physical space $x\in\R^2$, where the
concentration vector possesses a fixed, prescribed value
$u_*\in\R^2$. The curve $z_*(t)$ describes
a one-frequency motion on a circle  before the transition to meander
movement. After transition the curve $x_0=x_0(t)$ becomes an 
epicycle, as shown in Figure~\ref{Fig3.4},~\ref{f:meanderpics}. We refer to
this (nonunique) point as the {\em tip} of the spiral; see Section 5.2.4. 
Beyond the two-frequency meandering motion. We recall more complicated motions
of the spiral tip have been observed; see Section 5.2.7.

Explanations based on a curvature description of the wavetrain
dynamics in the spirit of Section~\ref{s:kin}, together with an
additional equation for the core of the spiral, were proposed
in~\cite{Mikhailov}. Barkley~\cite{Barkley92} was the first to suggest
simple Hopf bifurcation in the presence of Euclidean symmetry as the
mechanism which leads to meandering. Barkley later discovered
numerically that drift of the tip is related to a resonance between Hopf
frequency and rotation frequency of the spiral
wave~\cite{Barkley93}. The numerical and theoretical predictions have
been confirmed in careful experiments.  In particular, the
Hopf-typical square-root scaling of the amplitude of the instability
as a function of the parameter driving the instability has been
verified~\cite{Ouy}. 

The patterns in the farfield appear as prominent super-imposed
spiral-like regions, where the local wavelength is larger than the
average. The numerical computation of critical eigenfunctions reveals an
apparent dramatic change of the radial growth behavior of the eigenfunction
when the bifurcation parameter is varied close to criticality: before
instability, the amplitude of the eigenfunction appears to grow with
the distance from the center of rotation, whereas after instability, the
eigenfunction seems to be localized. Experimentally, the instability
appears to be  localized at the center of rotation in physical space. 
However, radial decay is very weak; see~\cite{Ouy}.

Spiral breakup is a phenomenologically different instability where, as
in the meander instability prominent super-patterns occur in the
farfield $|x|\to\infty$. This time, however, perturbations grow in
amplitude with distance from the center; see~\cite{BE,BO,OF,ZO}.
Before spiral breakup, the perturbations of the spiral wave again take
the form of superimposed spirals of compression and expansion of the
local wavelength of the wavetrains.  Instability finally leads to
visible collision of emitted pulses. Collision is generally referred
to as breakup and eventually leads to the formation of new spiral
cores, far away from the center of the primary spiral.The transition
to instability is not sharp and depends on the size of the domain. We
refer to this instability mechanism as {\em farfield breakup}.

Spiral breakup, i.e. collision of wavetrains, can also occur close to the
core region, $r\to 0$, of the primary spiral. After collision, new
spirals form close to the core of the primary spiral, which break up
immediately. This mechanism of instability leads to very incoherent
spatio-temporal states, with many small spiral domains;
see~\cite{AKW,BO}. We refer to this scenario of instability as {\em
core breakup}.

Both spiral breakup scenarios, farfield and core breakup, may occur as
instabilities of rigidly rotating {\em or} meandering spirals
\cite{ZO,BO}.

Aranson and Kramer~\cite{Kramer}, considering the complex
Ginzburg-Landau model equation, point out that the lack of a sharp
transition to instability in spiral breakup is caused by transport of
the wavetrains, rendering the instability convective: perturbations
increase in amplitude, but
decay at each fixed point in physical space.  
Sensitivity to instability can be increased by either further
increasing the bifurcation parameter, or, by increasing the size of
the domain. Baer et al. compare breakup instabilities with
instabilities of one-dimensional wavetrains emitted by a Dirichlet
source~\cite{BO}. Striking qualitative and quantitative similarity
suggests that spiral breakup is caused by an instability of the
wavetrains in the farfield.

We briefly mention three other instability phenomena without
attempting a complete explanation, here. The retracting wave
bifurcation~\cite{Mikhailov} manifests itself in a decrease of the
spiral frequency $\omega_*$ to zero. The radius of the circle $x_0(t)$
that a fixed point $u_0$ on the spiral profile describes diverges
until the spiral finally leaves the window of
observation. The instability is partially explained in~\cite{AshMel}
invoking a finite-dimensional reduction procedure.  Two-dimensional,
fingering-type instabilities of wavetrains, emitted by a spiral wave,
have been studied in~\cite{Panfilov}.  Period-doubling of a
homogeneous oscillation which induced complicated bifurcations of the
emitted wavetrains, has been observed in oscillatory
media~\cite{perioddoubling}.

We do not attempt  to list particular models giving rise to
the plethora of phenomena around spiral waves. Most models center around
two- or three-species reaction-diffusion equations with excitable or
oscillatory kinetics. A different line of investigations concentrates
on  the complex Ginzburg-Landau equation, where in particular the
wavetrain dynamics are amenable to explicit, algebraic
computations. 

Our emphasis is on  the general framework, assuming
existence of spiral waves without recurring to the precise
kinetics. Instability scenarios will be explained from certain
spectral assumptions on the linearization, rather than discussions of
oscillatory versus excitable kinetics.

As a general guideline,  we expect instabilities to be caused
either by essential spectrum or by point spectrum crossing the
imaginary axis. Essential spectrum crossing the axis will typically
not lead to an instability in bounded domains, where the absolute
spectrum, strictly to the left of the essential spectrum, decides upon
stability.

%\setcounter{equation}{0}
\subsection{Meandering spirals and Euclidean symmetry}
\label{s:meander}

In this section we consider a given rotating wave solution
$u=q_*(r,\varphi-\omega_* t)$ of a
reaction-diffusion system 
% 
\begin{eqnarray}u_t &=& D \Delta u+f(\mu,u). \label{3.3.1}\end{eqnarray}
We consider $u\in\mathr^N,$ a positive diagonal diffusion matrix $D,
x\in\mathr^m$, mostly $m=2$, and a scalar bifurcation parameter $\mu.$
For $f\in C^2,$ system (\ref{3.3.1}) fits into the abstract framework
of analytic semigroup theory on a Banach space $X$, \cite{Henry,pazy83},
but we prefer to formulate results in the more specific setting
(\ref{3.3.1}). Specifically we may think of $X=L^2,$ excluding the
Archimedean spirals of Chapter~\ref{s:3}, which do not decay in
amplitude for $|x|\rightarrow \infty.$ Suitable subspaces $X$ of
$BUC$, the space of bounded uniformly continuous functions, are
another possible choice. Although $BUC$ does accommodate Archimedean
spirals, these may exhibit critical continuous spectrum --- a
complication which we postpone to Section~\ref{s:3.3}.

\subsubsection{Euclidean equivariance}
Since the Laplacian $\Delta$ as well as the point evaluation nonlinearity
commute with translations $S\in\mathr^m$ and rotations $R\in SO(m)$ of
spatial profiles $u\in X$, we observe equivariance of the semiflow
generated by (\ref{3.3.1}) under the Euclidean symmetry group
% 
\begin{eqnarray} 
G = SE(m) = \{ (R,S);\; (R\in SO(m),\; S\in
\mathr^m\}. \label{3.3.2}
\end{eqnarray} 
The action on $g=(R,S)\in G$ on $u=u(x)$ is given explicitly by
% 
\begin{eqnarray}((R,S)u)(x) &:=& u((R,S)^{-1} x). \label{3.3.3}\end{eqnarray}
Note that the action on $x\in\mathr^m$ is affine,
% 
\begin{eqnarray}(R,S)x &:=& Rx + S, \label{3.3.4}\end{eqnarray}
whereas the action on $u\in X$ is linear. {\em Equivariance }of
(\ref{3.3.1}) under $G$ means that $u(t)\in X$ is a solution if, and
only if, $gu(t)\in X$ is a solution, for every fixed $g\in G.$ For a
general background of bifurcation theory in the presence of (mostly
compact) symmetry groups we refer to
\cite{cholau00,golsch85,gostsc88,vander82}. 

The group $G=SE(m)$ can be viewed as a Lie subgroup of the special
linear group $SL(m+1)$ of real $(m+1)\times (m+1)$ matrices with
determinant one, via the embedding
% 
\begin{eqnarray}\begin{array}{rcl} SE(m) &\rightarrow & SL(m+1)\\
(R,S) & \mapsto & \left(\begin{array}{cc} R & S\\0 &1 \end{array}\right)
\end{array}. \label{3.3.5}\end{eqnarray}
For example, $(R,S)^{-1}=(R^{-1}, -R^{-1}S).$ 

The embedding (\ref{3.3.5}) is also useful for practical computations
of the Lie algebra $se(m)$ and of the exponential map
$\exp:se(m)\rightarrow SE(m)$. For example ${\bf a}:=({\bf r,s})\in
se(m)$ is given by a skew symmetric matrix ${\bf r}\in so(m)$ and by
${\bf s}\in\mathr^m$ with explicit exponential
% 
\begin{eqnarray}\exp ({\bf r,s}) &=& (\exp {\bf r,\; r}^{-1}(\exp({\bf r})-{\rm id}){\bf s}).
\label{3.3.6}\end{eqnarray} 
For nonzero ${\bf r}\in so(2),$ we therefore see how any exponential
$\exp({\bf a}t)$ is conjugate to a pure rotation ${\bf s}=0,$ by a
suitable translation. Identifying ${\bf r}\in so(2)$ with the purely
imaginary number $\rmi \omega,$ we can also write rotations $R\in
SO(2)$ as $R=\exp ({\bf r}t)=\exp (\rmi \omega t).$ See \cite[Ch.
4]{FSSW} for many more examples.

In this abstract setting, a rotating wave solution $u(t)=u(t,\cdot)\in
X$ of (\ref{3.3.1}) takes the form
% 
\begin{eqnarray}
u (t) &=& \exp ({\bf a}_0 t) q_*, 
\label{3.3.7}\end{eqnarray}
for some $q_*\in X.$ The polar coordinate formulation (\ref{e-sppc})
of Section \ref{s:3.1.1} coincides with (\ref{3.3.7}) for ${\bf a}_0 =
({\bf r}_0,0)$ and ${\bf r}_0=\rmi \omega_*$. More generally, solutions
of the form (\ref{3.3.7}), with an arbitrary element ${\bf a}_0$ of
the Lie algebra ${\rm alg} (G)$ of $G$, are called {\em relative
equilibria:} the time orbit (\ref{3.3.7}) is entirely contained in the
group orbit $Gq_*$ and hence does not exhibit any shape
change. Travelling waves, where $G=(\R,+)$, are another simple
example; see Chapter~\ref{s:1u}. 

\subsubsection{Relative center manifolds}
We now describe the center-manifold reduction near relative equilibria
(\ref{3.3.7}) due to \cite{SSW,SSW1,SSW2}. We restrict to the slightly
simpler case of symmetry groups $G\leq SE(m)$. Fix the parameter
$\mu=\mu_0$. Assume the reaction kinetics $f=f(u)$ are smooth, $C^\infty$, and
fix $k\in\N$ arbitrary. We first pass to a ``rotating'' or ``comoving''
coordinate frame $\tilde{u}:=\exp(-{\bf a}_0t)u$, where $u(t)$ becomes
an equilibrium $\tilde{u}=:q_*$.  By $G$-equivariance, this
transformation leaves (\ref{3.3.1}) autonomous. Assume that the group
action is continuous on the relative equilibrium $q_*$. The central
assumption to Theorem~\ref{t-3.1}, below, is that the linearization at
the equilibria $Gq_*$ possess only point spectrum in $\{\Re\lambda\geq
-\delta\}\subset\C$, for some $\delta>0$. Let $W$ denote the
corresponding finite-dimensional generalized eigenspace.

%%th3.1
\begin{Theorem}\cite{SSW,SSW1,SSW2}\label{t-3.1}
Under the above assumptions there exists a $G$-invari\-ant
neighborhood $\cal{U}$ of the group orbit $Gq_*$ of the relative
equilibrium $q_*$ and a $G$-invariant $C^k$ {\em center manifold} ${\cal
M}\subseteq \cal{U}$ with differentiable proper $G$-action on ${\cal
M}.$ The $C^k$-manifold ${\cal M}$ is time-invariant within $\cal{U},$
and tangent to the center eigenspace $W$ at $q_*\in {\cal M}.$
Moreover ${\cal M}$ contains any solution $u(t),t\in\mathr,$ which
remains in $\cal{U}$ for all positive and negative times $t.$
\end{Theorem}


Since the center manifold ${\cal M}$ is based on a parameterization
over the group orbit $Gq_*$ of the relative equilibrium $q_*$, we call
${\cal M}$ a {\em relative center manifold}.  

Properness of the $G$-action on ${\cal M}$ is a technical property,
which asserts that the map
% 
\begin{eqnarray}
\begin{array}{rcl} G\times {\cal M} &\rightarrow & {\cal M} \times {\cal M}\\
(g,u) & \mapsto & (u,gu) \end{array} \label{3.3.9}
\end{eqnarray} 
is {\em proper}: images of closed subsets are closed and preimages of
points are compact. For example, consider the {\em isotropy} $H$ of $q_*\in {\cal
M}$:
% 
\begin{eqnarray}H=G_{q_*} := \{ h\in G;\; h{q_*}={q_*}\}. 
\label{3.3.10}\end{eqnarray}
The isotropy $H$ consists of the symmetries of the solution $q_*$ in
the group $G$. Note that isotropies are compact, for proper group
actions. Indeed $H\times \{u\}$ is the preimage of $({q_*},{q_*})$
under the proper map (\ref{3.3.9}), and hence compact.

We caution the reader that the eigenspace $W$ is not $G$-invariant, in
general. For example $G=SE(2)$ does not possess any nontrivial
finite-dimensional $G$-invariant subspace in $L^2.$ In fact, $g\in G$
will move $W$ to contain the tangent space $T_{gq_*}Gq_*$ which may
only intersect trivially with $W$. In particular, the center manifold
${\cal M}$ cannot be written as a graph of a globally $G$-equivariant
map over the tangent space $W.$ This situation differs markedly from
the case of compact symmetries $G$, where $X$ decomposes into
finite-dimensional $G$-invariant subspaces.

The proof of theorem \ref{t-3.1} first exploits finite-dimensionality of the center-eigenspace
in order to prove smoothness of the group action on spectral
projections associated with $W$ and on individual elements of $W$. This
allows for a smooth parameterization of the group orbit
$G(q_*+W)$. The main difficulty is that the group action on the normal
bundle, consisting of the strongly stable subspace, is not even strongly
continuous, in general. Still, the normal bundle is smooth, as a fiber
bundle, and discontinuity of the group action can be carefully
circumvented in the construction of the center manifold as a section to
the normal bundle. For details of the construction, we refer to
\cite{SSW2}.

\subsubsection{Palais coordinates}
Within the  finite-dimensional relative center manifold ${\cal M}$, we now
express the PDE dynamics (\ref{3.3.1}) by an ODE, rather explicitly,
using {\em Palais coordinates} $(g,v).$ See \cite{FSSW} for
details. We choose a local Palais section $V$ in ${\cal M}$,
transverse at ${q_*}$ to the group orbit $G{q_*}$. Then
% 
\begin{eqnarray}\begin{array}{rcl} G\times V &\rightarrow & {\cal M} \\
(g,v) & \mapsto & g({q_*}+v) \end{array} \label{3.3.11}\end{eqnarray}
identifies ${\cal M}$ as a principal fiber bundle over $G{q_*}$, with fiber
$V$ and structure group $H$. Indeed $(g,v)$ and $(g_0,v_0)$
parameterize the same point in ${\cal M}$ if, and only if,
% 
\begin{eqnarray}\begin{array}{rcl} g {q_*} & = & g_0{q_*},\quad {\rm\;and}\\
gv & = & g_0 v_0. \end{array} \label{3.3.12}\end{eqnarray} Therefore
$g=g_0h$, for some $h\in H,$ and thus $v_0=h v.$ Hence
% 
\begin{eqnarray} (g_0,v_0) &=& (gh^{-1}, hv). \label{3.3.13}\end{eqnarray}
With respect to this free action of the isotropy $H$ on $(g,v)\in
{\cal M}$, we indeed see that
% 
\begin{eqnarray}{\cal M} &=& (G\times V)/ H \label{3.3.14}\end{eqnarray}
is the orbit space of $H$. The action of $g_0\in G$ on ${\cal M}$ lifts
to $(g,v)\in G\times V$ canonically:
% 
\begin{eqnarray}g_0(g,v) &:=& (g_0 g,v) \label{3.3.15}\end{eqnarray}
in the above notation. Combining the actions (\ref{3.3.13}),
(\ref{3.3.15}) we see how $(g_0,h)\in G\times H$ act by
% 
\begin{eqnarray}(g,h)(g,v) &:=& (g_0 g h^{-1},hv) \label{3.3.16}\end{eqnarray}
on $(g,v)\in G\times V.$ We admit that our above presentation is simplistic
due to the suggestive, but strictly speaking illegitimate, use of the
notation $g({q_*}+v)$ for an element in the transverse slice $gV$ to the
group orbit $G{q_*}$ at $g{q_*}$. See \cite{FSSW} for full details.  An
alternative approach, focusing on the dynamics in the bundle ${\cal
M}$ directly, was developed earlier by \cite{krupa90} for compact groups
$G.$

We now lift the flow from the relative center manifold ${\cal M}$ to
the Palais coordinates $(g,v)\in G\times V.$ By equivariance
(\ref{3.3.15}) alone, this flow must take the following form.

%%th3.2
\begin{Theorem}\cite{FSSW}\label{t-3.2}
In Palais coordinates $(g,v)$, the $G$-equivariant flow (\ref{3.3.1})
in the relative center manifold ${\cal M}$ takes the skew product form
% 
\begin{eqnarray}\begin{array}{rcl} \dot{g} & = & g\mbf{a}(v) \\
\dot{v} & = & \Phi(v) \end{array} \label{3.3.17}\end{eqnarray} for
suitable vector fields $\Phi(v)$ on $V$ and $\mbf{a}(v)$ on the Lie
algebra ${\rm alg}(G).$ With respect to the compact isotropy $H$ of
${q_*}$ defined in (\ref{3.3.10}), these vector fields transform according
to
% 
\begin{eqnarray}\begin{array}{rcl} \Phi(hv) & = & h\Phi (v) \\
\mbf{a}(hv) & = & h\mbf{a}(v)h^{-1} \end{array} \label{3.3.18}\end{eqnarray}
\end{Theorem}

For analogues of Theorem \ref{t-3.2} for nonautonomous, periodically forced
systems see~\cite{SSW2}.  For bifurcations from relatively periodic
solutions, rather than relative equilibria, see \cite{SSW2,wulame01}.

The skew product form (\ref{3.3.17}) is easily interpreted. Points
$(g,v)\in G\times V$ with the same $v$-component mark points on the
same $G$-orbit in ${\cal M}$, by the $G$-action
(\ref{3.3.15}). Therefore $v(t)$ indicates the {\em shape} of the
original $x$-profile $u(t,\cdot)\in X$. The $G$-component $g(t)$, in
contrast, indicates {\em position} $S(t)$ and {\em phase} $R(t),$ at
least in the case $G=SE(m),\; g=(R,S),$ which is of primary interest
here.

\subsubsection{Spiral tip motion, Hopf meandering, and drift resonance}
To understand meandering spiral patterns, we rewrite (\ref{3.3.17}) explicitly in terms of
$g=(\rme^{\rmi\alpha},z)\in SE(2)$, with obvious complex notation for phase $R=\rme^{\rmi\alpha}$ and
position $S=z\in \mathc$. We immediately obtain the general form
% 
\begin{eqnarray}\begin{array}{rcl} \dot{\alpha} & = & \omega(v) \\
\dot{z} & = & \rme^{\rmi\alpha}\sigma(v) \\ \dot{v} & = &
\Phi(v). \end{array} \label{3.3.19}\end{eqnarray} We now argue that
the position component $z=z(t)$ can be interpreted as {\em tip
position} of a meandering spiral wave directly; see
\cite{fietur98}. Let
% 
\begin{eqnarray}z_* : {\cal M}\rightarrow \mathc \label{3.3.20} \end{eqnarray}
be any function which associates to a (nearly) spiral profile $u\in
{\cal M}\subseteq {\cal U}$ near $G{q_*}$ a position $z_*(u)\in\mathc
=\mathr^2$ of its ``tip''. Several such ``tip'' functions $z_*$ have
been considered; see Figure \ref{Fig3.3}. They all share the
equivariance property
% 
\begin{eqnarray}z_* (gu) &=& g z_*(u) \label{3.3.21} \end{eqnarray}
and hence lift to functions
$$z_*:G\times V\rightarrow \mathc$$ in Palais coordinates, which we
again denote by $z_*.$ A simple calculation then shows that
%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,7.)%
\centerline{\includegraphics[scale=0.6]{Fig/14.eps}}
\end{picture}%
\caption{``Tips'' of a spiral wave pattern in the BZ reaction: (a) core center, (b) maximum
curvature, (c) rotation center, (d) inflection point; courtesy of \cite{muezyk94}.} \label{Fig3.4}
\end{figure}
% 
\begin{eqnarray}\dot{z} &=& \rme^{\rmi\alpha} \sigma_*(v), \label{3.3.22} \end{eqnarray}
which takes the same form as the positional $z$-equation in
(\ref{3.3.18}). Only the precise form of the nonlinearity $\sigma$
depends on the precise form of the tip function $z_*.$ Discussing the
dynamics of skew product systems (\ref{3.3.19}), in general, will therefore
capture all the rather similar dynamics which are observed in the
experimental literature --- in spite of the varying definitions of a
``tip''. We therefore call $z(t)$ itself the tip of the spiral,
henceforth.

Returning to a parameter dependent version of the skew product
(\ref{3.3.19}), we give a simple example. Consider shape variables
$v\in\mathr^2$ undergoing Hopf bifurcation, due to a nontrivial pair
of imaginary eigenvalues of the linearization at ${q_*}$, in
corotating coordinates, crossing the imaginary axis transversely as
the parameter $\mu$ increases. Then the shape variable $v=v(t)$ becomes
periodic, say of Hopf frequency $\omega_{\rm H}.$ Dividing
(\ref{3.3.19}) by the nonzero Euler multiplier $\omega(v)$ does not
change the trajectory of the tip position $z\in\mathc$ and retains the
abstract form of (\ref{3.3.19}). We may therefore consider
$\omega(v)\equiv\omega_*$ where $\omega_*$ indicates the normalized
rotation frequency of the underlying rotating wave ${q_*}$, alias
$v\equiv 0.$

The tip motion $z(t)$ is then obtained by simple integration. Fourier expansion
% 
\begin{eqnarray}\sigma(v(t)) &=& {\displaystyle\sum_{k\in\MathZ}}
\sigma_k \rme^{\rmi k\omega_* t} 
\label{3.3.23}
\end{eqnarray} 
provides the explicit expression 
% 
\begin{eqnarray} 
z(t) &=& z_0 +\displaystyle{\int^t_0 \rme^{\rmi \omega_*
t'}}\sigma(v(t')) dt' = \nonumber \\ & =& z_0 +
\displaystyle{\sum_{k\in\MathZ}\frac{\sigma_k}{\rmi
(\omega_*+k\omega_{\rm H})}} (\rme^{\rmi (\omega_*+k \omega_{\rm
H})t}-1) \label{3.3.24}
\end{eqnarray} 
for Hopf frequencies $\omega_{\rm H}$ which are not an integer
fraction of the normalized rotation frequency $\omega_*.$ Clearly,
(\ref{3.3.24}) then describes a two-frequency quasi-periodic motion
% 
\begin{eqnarray}z(t) &=& \tilde{z}_0+ \tilde{\sigma}(t) \rme^{\rmi \omega_* t} \label{3.3.25}\end{eqnarray}
with $\tilde{\sigma}$ periodic of frequency $\omega_{\rm H}.$ For irrational
frequency ratios $\omega_*/\omega_{\rm H},$ the tip position $z(t)$ densely
fills an annulus around $\tilde{z}_0$ of inner and outer radius
$\min|\tilde{\sigma}|$ and $\max|\tilde{\sigma}|,$ respectively. At
integer {\em drift resonances}
%
\begin{eqnarray} 
k\omega_{\rm H} &=& \omega_* 
\label{3.3.25b}\end{eqnarray} 
the tip motion $z(t)$ typically becomes unbounded, due to a nonzero drift term $\sigma_{-k}t.$

Such meandering and drifting motions have been observed
experimentally, and numerically; see the discussion in
Section~\ref{s:3.4}. In view of work by \cite{mielke91} in the Hamiltonian
context, even the ``slinky'' spiral springs described by Love as far
back as 1892 \cite{love04}, can be considered as an example for
meandering. A first mathematical analysis of meandering spirals is due
to \cite{wulff96}.

\subsubsection{Relative normal forms}
With these interpretations in mind, we now proceed to sketch a normal
form analysis of skew products (\ref{3.3.17}), due to \cite{fietur98}. For
standard normal form procedures, going back essentially to Poincar\'e,
see the modern account in \cite{van}. For vector fields like the shape
dynamics $\dot{v}=\Phi(v)$, these procedures successively simplify
finite order terms in the Taylor expansion
% 
\begin{eqnarray}
\Phi (v) &=& M v+ \cdots \label{3.3.26}
\end{eqnarray}
near the equilibrium $\Phi(0)=0,$ by transformations
% 
\begin{eqnarray}
\tilde{v} = \Psi(v) = v+\cdots. \label{3.3.27}
\end{eqnarray}
The action of the transformations $\Psi$ on $\Phi$ is given explicitly
by the vector field pull-back
% 
\begin{eqnarray}\tilde{\Phi}(v) &=&
\Psi'(\Psi^{-1}(v))\Phi(\Psi^{-1}(v)). 
\label{3.3.28}\end{eqnarray}
Note linear dependence of $\tilde{\Phi}$ on $\Phi.$ The standard
normal form theorem asserts that nonlinear, finite order terms of
$\Phi(v)$ in the range of the linear map $\Phi\mapsto {\rm
ad}(M)\Phi$ can be eliminated, by suitable normal form
transformations $\Psi.$ Here
% 
\begin{eqnarray}
(({\rm ad} M)\Phi)(v) &:=& M\Phi(v) - \Phi'(v)Mv 
\label{3.3.29}\end{eqnarray}
preserves polynomial order. A normal form of $\Phi$ thus amounts to a
nonunique choice of a complement of ${\rm range}({\rm ad}(M)).$
Nonremovable terms, in this procedure, are frequently called {\em
resonant.}

For $G$-equivariant skew products (\ref{3.3.17}) we consider such a
normal form procedure as already applied to the shape equation
$\dot{v}=\Phi(v).$ The only remaining transformation then takes the
general form $\tilde{g}=g_0(g,v)$. The requirement of $G$-equivariance
(\ref{3.3.15}) enforces the particular form $\dot{g}=g\mbf{a}(v)$ in
(\ref{3.3.19}), which our normal form transformation $\Psi$ is supposed to
preserve. Therefore our normal form transformation has to be of the
form
% 
\begin{eqnarray}\tilde{g} &=& g g_0(v); \label{3.3.30}\end{eqnarray}
with $g_0(0)=0.$ Geometrically, such a transformation simply
corresponds to the choice of a different slice $V$ to the group action
in the center manifold ${\cal M}$; see (\ref{3.3.11}).


The induced transformation of the Lie algebra factor $\mbf{a}(v)$ is then given by
% 
\begin{eqnarray}\tilde{\mbf{a}}(v) &=& g_0^{-1}(v)\mbf{a}(v) g_0(v)+ g_0^{-1}(v)g'_0(v)\Phi (v)
\label{3.3.31}\end{eqnarray}
with $g_0^{-1}$ denoting the group-inverse of $g_0\in G$. Again, $\tilde{\mbf{a}}$ depends (affine)
line\-arly on $\mbf{a}$. 

To formulate our normal form result, in this setting, we introduce
some notation. Let $p$ enumerate the Jordan blocks of (not necessarily
distinct) eigenvalues $\lambda_p$ of $M=\Phi'(0)$, and let $1\leq q\leq
d_p$ enumerate a basis for each block. Then the shape variable $v$
possesses components $v_{pq}$ such that
% 
\begin{eqnarray}(Mv)_{pq} &=& \lambda_p v_{pq} + v_{p,q+1} \label{3.3.32}\end{eqnarray}
with the convention $v_{p,d_p+1}:=0.$ With integer nonnegative
multi-indices ${\bf k}=(k_{pq})_{pq}$ we expand
% 
\begin{eqnarray}\tilde{g} &=& \sum_{{\bf k}}\mbf{a}_{{\bf k}} v^{{\bf k}}, \label{3.3.33}\end{eqnarray}
with $\mbf{a}_0=\mbf{a}(0)$ unaffected by our transformation. We
further decompose each $\mbf{a}_{{\bf k}}$ spectrally by the adjoint
action
% 
\begin{eqnarray}\begin{array}{rcl} {\rm ad}(\mbf{a}_0) : & {\rm alg}(G) & \rightarrow {\rm alg}(G) \\
& \mbf{a} & \mapsto [\mbf{a}_0,\mbf{a}] \end{array} \label{3.3.34}\end{eqnarray} on the
Lie algebra ${\rm alg}(G).$ Let $\eta_j$ denote the distinct
eigenvalues of this action, and spectrally decompose $\mbf{a}_{{\bf
k}}=\sum_j\mbf{a}^j_{{\bf k}}$ accordingly.

{\em Resonant terms}, alias nonremovable terms, in our setting are
those monomials $\mbf{a}_{{\bf k}} v^{{\bf k}}$ in the expansion
(\ref{3.3.33}) which satisfy the {\em resonance condition}
% 
\begin{eqnarray}0=\eta_j+({\bf k},\lambda) := \eta_j+ \sum_{p,q} k_{pq} \lambda_p. \label{3.3.35}\end{eqnarray}
Note how this resonance condition captures an integer interaction
between the eigenvalues $\lambda_p$ of the linearized shape dynamics
$\dot{v}=Mv$ and the eigenvalues of the unperturbed, exponential group
dynamics $\dot{g}=g\mbf{a}_0$ along the relative equilibrium $v=0.$ We have
already encountered such a resonance, with $\eta=\rmi \omega_*$ and
$\lambda=\rmi \omega_{\rm H},$ as drift resonance of meandering spirals; see
(\ref{3.3.25b}) and (\ref{3.3.7}).

%%%th3.3
\begin{Theorem}\cite{fietur98}\label{t-3.3}
With the above assumptions and notations, there exists a normal form
transformation (\ref{3.3.30}) which preserves the skew product
structure (\ref{3.3.17}) and removes all nonresonant terms $\mbf{a}^j_{{\bf
k}}$ from $\mbf{a}(v)$, for any finite order $1\leq |{\bf k}|=\sum k_{pq}.$
The $H$-equivariance (\ref{3.3.13}), (\ref{3.3.18}) of Palais
coordinates can also be preserved. In the parameter-dependent case
$\mbf{a}=\mbf{a}(\mu,v),$ the same elimination holds with coefficients
$\mbf{a}_{{\bf k}}(\mu)$ and with resonance defined at $\mu=0,v=0.$
\end{Theorem}
We call any normal form reduction in the sense of Theorem \ref{t-3.3} {\em
relative normal form}.

\subsubsection{Relative Hopf resonance}

Instead of giving a proof of relative normal form Theorem \ref{t-3.3},
we illustrate some consequences for the Euclidean group $G=SE(2),$ and
for possible motions of spiral tips, by several examples. We begin
with standard Hopf bifurcation at $\mu=\mu_0=0\in\mathr,$ as was
already considered in (\ref{3.3.19}), (\ref{3.3.23}) --
(\ref{3.3.25b}). Normal form theory for Hopf bifurcation in
$\dot{v}=\Phi(\mu,v)$ implies $\Phi=\Phi_0(\mu,|v|^2)v,$ in complex
notation for $v\in\mathc.$ Therefore $|v|\equiv const.,$ for periodic
solutions $v(t).$ The Hopf eigenvalue is $\lambda=\rmi \omega_{\rm
H}\neq 0,$ of course. In the notation (\ref{3.3.2}) -- (\ref{3.3.6})
we have
% 
\begin{eqnarray}\mbf{a}_0=({\bf r}_0,{\bf s}_0) &=& (\rmi \omega_*,0) \label{3.3.36}\end{eqnarray}
for the rotating wave at $v=0.$ Therefore $\eta\in {\rm spec}\; \mbf{a}_0=\{\pm
\rmi \omega_*,0\}$ provides a resonance (\ref{3.3.35}) if, and only if,
% 
\begin{eqnarray}0 &=& -\omega_* + k\omega_{\rm H}. \label{3.3.37}\end{eqnarray}
This is precisely the condition for drift resonance encountered in (\ref{3.3.25b}) above. For
noninteger $\omega_*/ \omega_{\rm H}$ we can eliminate all $v$-terms in $\dot{g}=g
\mbf{a}(\mu,v),$ and the normal form
% 
\begin{eqnarray}\dot{g} &=& \tilde{g} \mbf{a}_0(\mu) \label{3.3.38}\end{eqnarray}
yields purely exponential solutions $\tilde{g}(t)$ with frequency
$\omega_*(\mu).$ Inverting the normal form transformation
(\ref{3.3.30}) then provides solutions
% 
\begin{eqnarray}g(t) &=& \tilde{g}(t) g_0(v(t))^{-1} \label{3.3.39}\end{eqnarray}
which are clearly quasiperiodic, with $g_0(v(t))^{-1}$ contributing the
Hopf-frequency $\omega_{\rm H}$ of the shape dynamics, and $\tilde{g}(t)$
contributing the rotation frequency $\omega_*$ of pure unperturbed
rotation. Since $g=(\rme^{\rmi \alpha},z)$, the tip motion $z(t)$ is also
quasiperiodic. Of course the same arguments apply to general
finite-dimensional Lie groups $G$. In particular, the case $G=SE(3)$
provides fascinating motions $z(t)\in\mathr^3$ of scroll rings (see
Chapter~\ref{c-4}) and vibrating arrows characterized by periodically
pulsating straight propagation and superimposed sidewards meandering;
see also \cite{FSSW}.

Similar calculations allow us to analyze $k:1$ drift resonance (\ref{3.3.37}) of the
frequencies $\omega_*:\omega_{\rm H}.$ The nonresonant normal form for the tip
motion $\tilde{z}(t)$ associated to the dynamics of $\tilde{g}(t)$ then becomes 
% 
\begin{eqnarray}\dot{\tilde{z}} &=& \rme^{\rmi \alpha}\sigma v^k \label{3.3.40}\end{eqnarray}
where $\sigma=\sigma(\mu_1,\mu_2)$ depends on
$\mu=(\mu_1,\mu_2)$ and $|v|^2.$ For simplicity, we
normalize $\mu_2=|v|^2 \geq 0$ as the Hopf bifurcation parameter
and let $\mu_1$ measure the detuning of the linearized Hopf
frequency $\omega_{\rm H}(\mu_1)=\omega_{\rm H}(0)+\mu_1$ at the
bifurcation line $\mu_1\in\mathr,\;\mu_2=0,\;v=0.$ Then
(\ref{3.3.40}) implies
% 
\begin{eqnarray}\tilde{z} (t) &=& c-\sigma\mu_2^{k/2} (\rme^{-ik\mu_1 t}-1)/ ik\mu_1. 
\label{3.3.41}\end{eqnarray}
For the tip motion $z(t)$ in original coordinates, we have to account
for the transformation (\ref{3.3.30}) by
$g_0(v)=(\rme^{\rmi \alpha_0(v)},z_0(v))$ of period (nearly) $2\pi/\omega_{\rm H}.$
In total we obtain a meandering tip motion of $z(t)$ along large
circles of radius proportional to
% 
\begin{eqnarray}\mu_2^{k/2} / |\mu_1|, \label{3.3.42}\end{eqnarray}
for $\mu_2>0,$ which blow up at the drift resonance $\mu_1=0.$
Strictly speaking we have suppressed terms in (\ref{3.3.40}) which may
not be in normal form. These terms however can be assumed to be
perturbations of arbitrarily high finite order.

The superposition $g(t)=\tilde{g}(t) g_0(v(t))^{-1}$ in (\ref{3.3.39})
also leads to curious tip motions right at bifurcation
$\mu=\mu_0=0,$ in these resonant cases. In nondegenerate
normal form and for stable $v\equiv 0,$ the shape component $v(t)$
converges to zero with $|v(t)|\sim t^{-1/2}$ and asymptotic phase
frequency $\omega_{\rm H}.$ For the tip motion $\tilde{z}(t)$ in normal form
coordinates, (\ref{3.3.40}) provides a constant limit
% 
\begin{eqnarray}\tilde{z}_\infty &=& \lim_{t\rightarrow \infty} z(t), \label{3.3.43}\end{eqnarray}
for orders $k\geq 3$ of the integer drift resonance
$k=\omega_*/\omega_{\rm H}.$ In terms of the true tip motion $z(t)$ this
corresponds to a rotating wave limit, because $\tilde{z}_\infty$ can
be eliminated by conjugation in $G=SE(2)$.

For $k=\omega_*:\omega_{\rm H}=2:1,$ however, $\tilde{z}(t)$ converges to a circular motion
% 
\begin{eqnarray}\tilde{z}(t) &\sim & c_0+ c_1 \exp(-i\tau(t)) \label{3.3.44}\end{eqnarray}
with constants $c_0,c_1$, asymptotically logarithmic phase $\tau(t)$,
and decreasing angular velocity $\dot{\tau}(t)\sim 1/t.$ The tip
motion $z(t)$ then corresponds to a meander with constant speed
epicycles along a slowing base circle.

The 1:1 drift resonance $\omega_*=\omega_{\rm H},$ finally, gives rise to an
unbounded motion (\ref{3.3.44}) with logarithmic phase and nonconstant
$c_1 \sim \sqrt{t}.$ All these results follow from just the normal
form equation (\ref{3.3.40}).

Clearly the analysis of relative mode interactions in the shape
variable $v$ becomes feasible now.  For an example involving Hopf-Hopf
mode interactions of two complex eigenvalues $\rmi \omega_1,\rmi
\omega_2$ see \cite{asmeni01}.

\subsubsection{Relative Takens-Bogdanov bifurcation}
The case of nilpotent linearization $M=\Phi'(0)$ in the shape
dynamics $\dot{v}=\Phi(v),$ with trivial (relative) equilibrium
$v=0$ contributes only eigenvalues $\lambda=0$ to the resonance condition
(\ref{3.3.35}). In normal form it turns out that
% 
\begin{eqnarray}\dot{\tilde{z}} &= & 0, \label{3.3.45}\end{eqnarray}
because all terms in the translational $\tilde{z}$-equation are
nonresonant. Indeed $\eta=\rmi \omega_*$ is generated only by the rotation
frequency $\omega_*$ of the reference rotating wave; the zero
eigenvalue of ${\rm ad}(\mbf{a}_0)$ does not appear in the component
$\tilde{z}.$ In particular, any drift terms vanish beyond finite
order.

A prime example, not observed experimentally so far, are the effects
of a relative Takens-Bogdanov bifurcation in the shape variable
$v\in\mathr^2.$ For classical analysis see
\cite{takens74,bogdan81a,bogdan81b,guchol83}. From (\ref{3.3.19}) we
recall the tip motion
% 
\begin{eqnarray}\dot{z} & = & \rme^{\rmi \omega_*t}\sigma(v(t)), 
\label{3.3.46}\end{eqnarray}
again after rescaling time so that $\dot{\alpha}\equiv \omega_*.$ If
$v(t)$ is homoclinic to a hyperbolic relative equilibrium $v_\infty,$
the resulting tip motion $z(t)$ satisfies
% 
\begin{eqnarray}z^h=\displaystyle{[z(t)]^{+\infty}_{-\infty}} & = & \int^{+\infty}_{-\infty}
\rme^{\rmi (\omega_*t+\alpha_0)}\sigma (v(t))dt. \label{3.3.47}\end{eqnarray}
The integral converges if $\sigma(v_\infty)=0$ at the rotating wave
$v_\infty.$ Again this can be assumed after a conjugation in the group
$G=SE(2)$. By hyperbolicity this implies exponential convergence
$\sigma(v(t))\rightarrow 0,$ for $t\rightarrow\pm\infty.$ The
homoclinic shape dynamics $\Gamma=\{v(t);\;t\in\mathr\}$ thus typically
implies heteroclinic tip dynamics, with heteroclinic tip shift
$z^h\neq 0.$

Next suppose that the homoclinic orbit $\Gamma\subseteq \mathr^2$ is
attracting from inside, due to a negative saddle quantity
$\lambda_1+\lambda_2< 0$ of the eigenvalues $\lambda_1<0<\lambda_2$ at its equilibrium
$v=v_{\infty}.$ Let $v(t)$ converge to $\Gamma$ from the inside. Then
the passage times $t_k$ near $v=v_\infty$ grow geometrically like
% 
\begin{eqnarray}t_k \sim c_1 \nu^k, \label{3.3.48}\end{eqnarray}
with $\nu=|\lambda_1/\lambda_2|>1.$ Therefore the respective tip positions behave like
% 
\begin{eqnarray}z(t_{n+n_0}) \sim z(t_{n_0}) + \sum^{n}_{k=n_0+1} z^h\cdot \exp(\rmi \omega_* c_1\nu^k).
\label{3.3.49}\end{eqnarray}
With respect to Lebesgue measure on $\omega_*,$ this trigonometric
power sum defines a random walk of $z(t_n)$ which can be rescaled to a
Brownian motion of $z$. Indeed the increments are then only weakly
coupled, for $k\rightarrow \infty.$ See \cite{BePh96} and, for explicit
sample plots, Figure \ref{Fig3.5}, with $\tilde{\omega}:=\omega_* c_1$.
%
%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,8.)%
\centerline{\includegraphics[scale=1.0]{Fig/15.eps}}
\end{picture}
\caption{Sample paths of Brownian tip motions due to asymptotically homoclinic shape dynamics.} \label{Fig3.5}
\end{figure}

We caution our reader on two points. First, the waiting times for each
step of our random heteroclinic tip walk increase exponentially at
rate $\log \nu >0$, with time.  Observation of such a random walk may 
therefore take a {\em very} long time. Second, although the
heteroclinic tip shift $z^h$ in (\ref{3.3.47}) will typically be
nonzero along homoclinic orbits $\Gamma,$ it will be small beyond
finite order in the Takens-Bogdanov case. See (\ref{3.3.45}). Indeed
the rotating wave phase $\exp(i\alpha(t))=\exp(\rmi \omega_*t)$ enters
as a rapid forcing into the slow dynamics generated by the zero
eigenvalues of the shape dynamics.  In spite of these subtleties, the
example shows clearly how a planar --- and therefore nonchaotic --
Poincar{\'e}-Bendixson type shape dynamics $v$ can cause complicated
tip motions in the group component $\dot{g}=ga(v)$ of the skew product
flow (\ref{3.3.17}).

\subsection{Spectra of spiral waves}
\label{s:3.3}
A major problem concerns a proper understanding of stability features
of spiral waves. Even the notion of spectral stability turns out to be
a very delicate problem. By means of a formal asymptotic matching
procedure, Hagan~\cite{Hagan} showed that spiral waves in the complex
Ginzburg-Landau equation (\ref{e:cgl}) are likely to be stable. We
give here a conceptual approach to the stability problem. We first
characterize essential and absolute spectra of the spiral wave in
terms of spectra of emitted wavetrains. We then describe the farfield
shape of eigenfunctions to eigenvalues in the point spectrum in terms
of the complex dispersion relation of the wavetrains.

\subsubsection{The eigenvalue problem for spiral waves: core versus
farfield} 
\label{s:3.3.1}
As already pointed out in the discussion after Definition~\ref{d:sw} a
variant of Proposition~\ref{p:ivc} holds for spiral waves. Fredholm
indices are again determined by group velocities on Fredholm
borders. In order to make a precise statement, we consider the
eigenvalue problem for the linearization ${\cal L}_*$ at an
Archimedean spiral $q_*(r,\psi)$. In corotating polar coordinates,
as defined in~(\ref{e:splin}), the eigenvalue problem $({\cal
L}_*-\lambda)U=0$ becomes
\begin{eqnarray}\label{e:linrad}
U_r&=&V\\ V_r&=&-\frac{1}{r} V - \frac{1}{r^2}\partial_{\psi\psi}U -
D^{-1}\left(-\omega_*\partial_\psi U + f'(q_*(r,\psi))U - \lambda
U\right).\nonumber
\end{eqnarray}
As a first approach to the eigenvalue problem, we try to detect
nontrivial bounded solutions of (\ref{e:linrad}). An analogous
consideration for the (formally) adjoint will then provide 
complete information about Fredholm properties such as the dimension
of the kernel and the dimension of the cokernel.

The idea is to construct stable and unstable subspaces, containing
solutions which are bounded in the farfield $r\to \infty $ and in the
core $r\to 0$, respectively. From the dimension of these subspaces, we
then read off the typical dimension of their intersection, which
yields the dimension of the kernel. The dimension of the complement of
their sum provides us with the dimension of the cokernel.

The core limit $r\to 0$ is again studied in the rescaled radial time
$\tau=\log(r)$:
\begin{eqnarray}\label{e:linradtau}
U_\tau &=&W\\
W_\tau &=& - \partial_{\psi\psi}U - \rme^{2\tau} D^{-1}\left(-\omega_*\partial_\psi U + f'(q_*(r,\psi))U - \lambda U\right).\nonumber 
\end{eqnarray}
In the limit $\tau=-\infty$, corresponding to $r=0$, we formally
obtain the autonomous, $\lambda$-independent system
\begin{equation}\label{e:cota}
U_\tau=V,\quad V_\tau = -\partial_{\psi\psi}U.
\end{equation}
The equation is readily solved by angular Fourier decomposition. We
find solutions to the spatial  eigenvalues $\nu=\pm \ell$,
\[
U(\tau,\psi)=U^\ell_+\rme^{\rmi\ell\psi +\ell \tau},\qquad  
U(\tau,\psi)=U^\ell_-\rme^{\rmi\ell\psi -\ell \tau},
\]
for $\ell\not=0$ and
\[
U(\tau,\psi)=U^0_+,\qquad  
U(\tau,\psi)=\tau U^0_-,
\]
for the constant Fourier mode $\ell=0$. The vectors
$U^\ell_\pm\in\C^N$ are arbitrary. The condition $U_-^\ell=0$ for all
$\ell$ defines the (infinite-dimensional) {\em unstable subspace}
$E_-^{\rm u}$ of solutions which stay bounded in the limit $\tau\to
-\infty$. Note that these asymptotic eigenspaces are independent of
the spectral parameter $\lambda$. A generalization of the concept of
exponential dichotomies from ordinary differential
equations~\cite{Palmer,Coppel} to the ill-posed, elliptic problem
(\ref{e:linradtau}) allows us to extend the unstable subspaces to the
non-autonomous system (\ref{e:linradtau}) as a $\tau$-dependent family
of subspaces $E_-^{\rm u}(\tau;\lambda)$; see ~\cite{PSS,Sspiralex}.

In the farfield limit $r\to\infty$, we formally obtain
\begin{eqnarray}\label{e:linradinf}
U_r&=&V\\
V_r&=&- D^{-1}(-\omega_*\partial_\psi U 
+ f'(q_{\infty}(r-\frac{\psi}{k_{\infty}}))U - \lambda 
U). \nonumber 
\end{eqnarray}
If we formally replace $\psi/\omega_*\mapsto t$, $r\mapsto x$, and
$U\mapsto U\rme^{-\lambda t}$, we obtain the linearization about the
one-dimensional wavetrains
\begin{equation}\label{e:linwt}
U_t=DU_{xx}+ f'(q_{\infty}(x-\frac{\omega}{k_{\infty}}t)U).
\end{equation}
We have already discussed this equation in a comoving frame
$\xi=x-\frac{\omega}{k_{\infty}}t$; see Section~\ref{s:1.2},
(\ref{e-fl}). 

Analogously to the unstable subspace $E_-^{\rm u}$ in the core region,
we are going to determine the stable subspace $E_+^{\rm s}$ in the
farfield equation (\ref{e:linradinf}). Since the farfield system is
periodic in spatial ``time'' $r$, Floquet theory
(\ref{e-flth})-(\ref{e:drper}) predicts that the stable subspaces are
``time''-dependent and exponential growth or decay is governed by
Floquet exponents $\nu$. In the next section, we determine these
Floquet exponents, depending on the spectral parameter $\lambda$.

\subsubsection{Spatial Floquet theory and the dispersion relation of
wavetrains} 
\label{s:3.3.2}
Our goal in this section is a characterization of exponential radial
decay and growth properties of solutions to~(\ref{e:linradinf}). The
procedure is strongly reminiscent of Floquet theory
(\ref{e-flth})-(\ref{e:drper}) in Section~\ref{s:1.2}, but the
additional dependence on the angle $\psi$ introduces some
complications.

To eliminate explicit dependence on the angle $\psi$
in~(\ref{e:linradinf}), we exploit the Archimedean shape of the spiral
by introducing ``Archimedean coordinates''. We replace $\psi$ by
$\xi=r-(\psi/k_{\infty})$, and find
\begin{eqnarray}\label{e:linradinfco}
U_r&=& -\partial_\xi U + V\\
V_r&=& -\partial_\xi V- D^{-1}(\frac{\omega_*}{k_{\infty}}\partial_\xi U 
+ f'(q_{\infty}(\xi))U - \lambda 
U). \nonumber 
\end{eqnarray}
Dependence of $q_\infty$ on $\xi$ is $2\pi/k_\infty$-periodic.  The
equation is now autonomous in $r$, and solutions are of the general
form $U=\rme^{-\nu r}\hat{U}(\xi)$, where $\hat{U}$ solves the
boundary-value problem
\begin{equation}\label{e:evas0}
D(\frac{\rmd}{\rmd\xi}-\nu)^2\hat{U}+\frac{\omega_*}{k_{\infty}}
\frac{\rmd}{\rmd\xi} \hat{U} + f'(q_\infty(\xi))\hat{U} =
\lambda\hat{U},
\end{equation} 
with periodic boundary conditions on $[0,2\pi/k_{\infty}]$. Note that
(\ref{e:evas0}) is equivalent to the boundary-value
problem~(\ref{e:cglin0}) for the dispersion relation of the
wavetrains. To simplify (\ref{e:evas0}) further, we introduce
$\tilde{U}(\xi):=\rme^{\nu\xi}\hat{U}(\xi)$. This yields
\begin{equation}\label{e:evas}
D\tilde{U}_{\xi\xi} + \frac{\omega_*}{k_{\infty}} \tilde{U}_\xi +  f'(q_\infty(\xi))\tilde{U} = \tilde{\lambda}\tilde{U}
,\end{equation}
with $\tilde{\lambda}:=\lambda-(\omega_*\nu/k_{\infty})$. Boundary
conditions are now of Floquet type
\begin{equation}\label{e:bc}
\tilde{U}(0)=\rme^{2\pi\nu/k_{\infty}}\tilde{U}(2\pi/k_{\infty})
\end{equation}
induced from the $2\pi/k_{\infty}$-periodic boundary conditions for
$\hat{U}$. 

The dimension of the stable subspace $E_+^{\rm s}$ for
(\ref{e:linradinf}) changes whenever a Floquet exponent $\nu$ crosses
the imaginary axis, as $\lambda$ is varied. We therefore focus on
solutions to (\ref{e:evas}), (\ref{e:bc}), with $\nu=\rmi k$, purely
imaginary, first. In analogy to the one-dimensional discussion in
Section~\ref{s:1.2}, we call $k$ the {\em Bloch wavenumber}.

Nontrivial solutions to the boundary-value problem
(\ref{e:evas0}),(\ref{e:bc}) can be found from the dispersion relation
\begin{equation}\label{e:drper2}
d_{\rm co}(\tilde{\lambda},\nu) ={\rm
det}\,(\Phi(\tilde{\lambda})-\rme^{2\pi\nu/k_\infty })= 0,
\end{equation}
where $\Phi(\lambda)$ denotes the period-$2\pi/k_\infty$ map to
(\ref{e:evas}), rewritten as a first-order differential equation; see
(\ref{e:drper}). Substituting
$\tilde{\lambda}=\lambda-(\omega_*\nu/k_{\infty})$ we obtain 
\begin{equation}\label{e:drper3}
d_{\rm st}(\tilde{\lambda},\nu) ={\rm
det}\,(\Phi(\lambda-(\omega_*\nu/k_{\infty}))-\rme^{2\pi\nu/k_\infty })= 0.
\end{equation}
If we fix $\Im\nu\in[0,k_\infty)$, we find $N$ solution
$\lambda^\ell(\nu)$. The Floquet shift $\nu\mapsto\nu+\rmi k_\infty$
gives us the vertically shifted spectral curves 
\begin{equation}\label{e:flsp}
\lambda^\ell(\nu)+\rmi\omega_*\Z.
\end{equation}
Note that the associated group velocity
\begin{equation}\label{e:cg3}
c_{{\rm g},j}( \rmi k):=\Im\left(\frac{\rmd \lambda_j(\rmi  k)}{\rmd
 k}\right)=\Im\left(\frac{\rmd 
\tilde\lambda_j(\rmi  k)}{\rmd  k}\right) +  c_{\infty},
\end{equation}
corresponds precisely to our previous definition (\ref{e:cglin}) of
the group velocity of the wavetrains.

Restricting to $\nu=\rmi k$, $k\in\R$, we obtain the spectral curves
$\Gamma^\ell = \{\lambda^\ell(\rmi k);\;k\in\R\}$. We emphasize that
together with a fixed spectral curve $\Gamma^\ell$, there is the
countable collection of vertically shifted (not necessarily disjoint)
curves $\Gamma^\ell+\rmi\omega_*\Z$. Comparing (\ref{e:linwt}) with
(\ref{e:evas}), we find that these curves consist of the continua of
Floquet exponents for the linearized period map for the
one-dimensional wavetrains.

For an illustration, we deviate from the general setup and discuss a
specific example. Assume that the wavenumber $k_\infty$ of the emitted wavetrains
is small. More specifically, assume that on their very large interval of
periodicity, $\xi\in[0,2\pi/k_\infty]$, the wavetrains resemble a
pulse solution $q_*(\xi)$, $|q_*(\xi)|\to 0$ for $|\xi|\to\infty$. We
then refer to the wavetrains as pulse trains. Spectral properties of
pulsetrains are determined by interaction of individual pulses, as we
shall explain next. In particular, we can compute the Floquet spectrum
of the pulse trains from certain geometric properties of a single
pulse, alone. First consider the pulsetrains and the pulse in a
comoving frame. Then the spectrum of the pulsetrains converges to the
spectrum of the single pulse~\cite{SSperiodic}. Close to the
(critical) zero-eigenvalue of the pulse with translation eigenfunction
$q'_*(\xi)$, the spectrum of the pulsetrains consists of a small
circle of essential spectrum
\[
\tilde\lambda(\nu)=a(k_\infty)(\rme^{2\pi\nu/k_\infty}-1)+
\rmo(|a(k_\infty)|),\quad \nu=\rmi k\in\rmi\R, ,
\] 
see Figure~\ref{f:specweak}(a). The coefficient $a$ is given as
$a(k_\infty)=M\rme^{-\delta/k_\infty}$. The constant $\delta>0$ is the
weakest exponential decay rate at the tails of the individual
pulse. The coefficient $M$ measures the (signed) speed of crossing of
stable and unstable manifolds of the origin in the travelling-wave
equation 
\[
Dq''+cq' + f(q)=0,
\]
when the speed $c$ is varied. Note that $a(k_\infty)\to 0$, exponentially, for
$k_\infty\to 0$. This reflects that the interaction strength between individual pulses decreases exponentially with their distance.

Spectral stability corresponds to $a(k_\infty)<0$. The crucial
coefficient $M$ turns out to be positive in the classical FitzHugh-Nagumo equation.

In a comoving coordinate frame of speed $c_{\infty}$, we have to add
the speed of the frame to the group velocity and find
\[
\lambda(\nu)=a(k)(\rme^{2\pi\nu/k_\infty}-1)+ c_{\infty}\nu +
\rmo(|a(k)|),\quad \nu=\rmi k\in\rmi\R 
;\]
see Figure~\ref{f:specweak}(b). The result is a vertically periodic
epicycloid, winding along the imaginary axis, with tangencies at
$\rmi\omega_*\Z$. 

\begin{figure}[!ht] 
\setlength{\unitlength}{1cm}% 
\begin{picture}(13.5,3.5)%
\centerline{\includegraphics[scale=0.7]{Fig/pulse-train-st-mo.eps}}
\end{picture}%
\caption[]{The spectrum of wavetrains of weakly interacting pulses in the steady frame (a), and the Floquet exponents in
the comoving frame (b).}   
\label{f:specweak}  
\end{figure} 

\subsubsection{Relative Morse indices and essential spectra of spiral waves} 
\label{s:3.3.3} 
In the spirit of Proposition~\ref{p:ivc}, the oriented curves
$\Gamma^\ell$ determine the Fredholm index of the linearization about a
spiral wave, see (\ref{e:linrad}). However, Proposition~\ref{p:fredi}
has to be reformulated since both Morse indices $i_\pm$ at
$\tau=-\infty$ (alias $r=0$) and $r=\infty$ are
infinite. Proposition~\ref{p:fredi} remains valid if both Morse
indices $i_\pm$ are normalized to finite numbers, the {\em relative
Morse indices}, which then enter formula (\ref{e:fredi}) for the
Fredholm index; see~\cite{SSmathnach} for details. A similar idea has
been used repeatedly in the construction of Floer homology in various
variational problems; see for example~\cite{RS95}. 

Since the asymptotic equation (\ref{e:cota}) in the core region,
$\tau\to-\infty$, does not depend on $\lambda$, we can fix the Morse
index in $-\infty$ and (arbitrarily) normalize to $i_-=0$. We show
next, how to define $i_+(\lambda)=:i_{\rm Morse}(\lambda)$, such that
Proposition~\ref{p:ivc} carries over to to linearizations at spiral
waves: the Fredholm index of ${\cal L}_*$ is again given by the
difference of the Morse indices 
\begin{equation}\label{e:spfm}
i({\cal L}_*-\lambda)=i_--i_+.
\end{equation} 

We solve the dispersion relation (\ref{e:drper3}) for the radial
Floquet exponent $\nu$ as a function of $\lambda$. For fixed
$\lambda$, there exists an infinite number of spatial Floquet
exponents $\nu_j$, with $0\leq\Im\nu<2\pi/k_\infty$, which solve
(\ref{e:drper3}). Just like in the example of the heat
equation~(\ref{e:heat}), the real parts of the $\nu$ are unbounded on
the positive and negative axis. Instead of counting unstable
eigenvalues, we therefore count differences between numbers of
unstable eigenvalues in different regions of the complex plane
$\lambda\in\C$.

Consider the (open) complement of the spectral curves in the
complex plane
\[
{\cal U}=\C\setminus\bigcup_{\ell=1}^N(\Gamma^\ell+\rmi\omega_*\Z).
\]
For all $\lambda\in{\cal U}$, the asymptotic equation in the farfield,
(\ref{e:linradinf}) is hyperbolic, that is, there are no purely
imaginary solutions $\nu=\rmi k$ to the dispersion relation
(\ref{e:drper3}). In consequence, the Morse index $i_{\rm Morse}$ is
constant on connected components of ${\cal U}$. 

In the connected component ${\cal U}_\infty$ of ${\cal U}$ containing
$\Re\lambda\to +\infty$, the Fredholm index is zero. Indeed, the
linearization ${\cal L}_*-\lambda$ is invertible, there, since the
parabolic reaction-diffusion system is well-posed. In view of
(\ref{e:spfm}), we define $i_{\rm Morse}(\lambda)\equiv 0$ in ${\cal
U}_\infty$.  

If we now move $\lambda$ from one connected component of ${\cal U}$ to
another, we cross a spectral curve
$\Gamma^\ell+\rmi\ell'\omega_*$. For
$\lambda\in\Gamma^\ell+\rmi\ell'\omega_*$, a Floquet exponent $\nu$ is
located on the imaginary axis and hyperbolicity
fails. The Morse index of ${\cal L}_*-\lambda$ changes by one when $\lambda$ is varied across
this single spectral curve. In order to determine the direction of
change, we orient the curves $\Gamma^\ell$ such that vertical slope
corresponds to positive group velocity, similarly to
Proposition~\ref{p:ivc}. With this orientation, the Floquet exponent
$-\nu$ (recall the Ansatz $U\sim\exp(-\nu r)$) moves from $\Re\nu<0$
to $\Re\nu>0$ when $\lambda$ crosses the spectral curve from left to
right; see also Lemma~\ref{l:cgweight}. In particular, the (relative)
number of Floquet exponents in $\Re\nu<0$ decreases by one each time
we cross spectral curves from left to right. We call this (relative)
number of Floquet exponents in $\Re\nu<0$ the {\em relative Morse index
$i_{\rm Morse}.$} This index is now defined in ${\cal U}$ via a
$\lambda$-homotopy from ${\cal U}_\infty$ to any point in ${\cal U}$,
counting signed crossings of the oriented curves $\Gamma^\ell +i\ell' \omega_* $ during the
homotopy.

\begin{Proposition}\label{p:fredisp}
The operator ${\cal L}_*-\lambda$, defined in (\ref{e:splin}),
considered as an unbounded operator on $L^2(\R^2)$ is Fredholm if, and
only if $\lambda\not\in \Gamma^\ell$, that is, if $\lambda$ belongs to
the complement ${\cal U}$ of the set of Floquet exponents of the emitted
wavetrains. The Fredholm index is given by
\[
i({\cal L}_*-\lambda)=-i_{\rm Morse}(\lambda),
\] 
where the relative Morse index $i_{\rm Morse}$ was introduced
above. In particular, the Fredholm index increases by one when
crossing any of the oriented Fredholm borders
$\Gamma^\ell+\rmi\ell'\omega_*$ from left to right.
\end{Proposition}
In the remainder of this section, we are going to derive a ``typical''
picture of the critical spectrum of a spiral wave.  First notice that
for $\lambda=0$, the derivative of the wave train $U=q_{\infty}'$
provides a trivial solution to~(\ref{e:evas}--\ref{e:bc}) with
$\nu=0$. If we assume that $\lambda=0$ is simple as a root of the
dispersion relation (\ref{e:drper3}), then there is a unique spectral
curve $\Gamma^0=\lambda^0(\rmi k)$, contained in the essential
spectrum, that touches the imaginary axis in $\lambda=0$. Since the
positive group velocity of the emitted wavetrain does not vanish, the
curve $\Gamma^0$ is not tangent to the real axis, see
Definition~\ref{d:sw}. Because the spectrum of ${\cal L}_*$ is invariant
under complex conjugation, the curve $\Gamma^0$ necessarily possesses
a vertical tangent at $\lambda=0$.  Typically, the tangency will be
quadratic. If the wave trains are spectrally stable, $\Gamma^0$ is the
rightmost Fredholm border in the complex plane. In $\Re\lambda>0$, the
Fredholm index of ${\cal L}_*$ therefore is 0. Since the group
velocity is positive, the region $\Re\lambda>0$ is to the right of the
Fredholm border $\Gamma^0$.  To the left of this curve $\Gamma^0$, in
$\Re\lambda<0$, the Fredholm index of the linearization is $-1$, by
Proposition~\ref{p:fredisp}. See Figure~\ref{f:spiralfred} for an
illustration of the resulting spectral picture.

Positive group velocity was encoded in Definition \ref{d:sw} by means
of exponential weights. Consider the linearization in $L^2_\eta,$
(\ref{e:spexp}), with radial weight $\eta<0$ close to zero. Then the
weight pushes the critical spectral curve $\Gamma$ into the stable,
left complex half plane. In other words, the exponentially decaying
weight penalizes the outward transport; see Section~\ref{s:1.2} and
Figure~\ref{f:spiralfred}. Note that translation
$(\partial_{x_1}+\rmi\partial_{x_2})q_*$ and angular derivative
$\partial_{\psi}q_*$ are bounded. They therefore contribute to the
kernel in the space $L^2_\eta$, which allows for exponential growth. 

\begin{figure}[!h]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,4.5)%
{\raisebox{-0.2cm}{\centerline{\includegraphics[scale=0.55]{Fig/spiralfred.eps}}}}
\end{picture}%
\caption[]{Typical spectrum of the linearization about a spiral wave
with Fredholm indices of the linearization in $L^2$ and in $L^2_\eta$,
for $eta<0$.} 
\label{f:spiralfred}  
\end{figure} 

Along the imaginary axis, the spectral picture is periodic with period
$\rmi\omega_*$. The essential spectrum touches the imaginary axis at
all points $\rmi\omega_*\Z$. Since the wavetrains emitted from the
center have a positive group velocity in the radial direction, this
implies that the Fredholm index of ${\cal L}_*-\lambda$ is negative, $i({\cal L}_*-\lambda)= -1$, for $\lambda$
to the left of all these critical spectral curves $\lambda^{\ell'}(\rmi k)$,
$\lambda^{\ell'}(0) =\rmi\ell'\omega_*$, .

The abundance of critical spectrum, in the form of infinitely many
tangencies of continuous spectrum at the imaginary axis, seems to
suggest that spiral waves are extremely sensitive to
perturbations. However, spiral waves in experiments appear to be very
stable objects in many systems! In the next section, we will find a
partial explanation for this apparent contradiction.


\subsubsection{Absolute spectra of spiral waves}
\label{s:abssp}
We have seen in Section~\ref{s:2} how stability properties of waves on
bounded domains are captured by the absolute spectrum rather than the
essential spectrum. In this section, we carry out the analogous
construction to define the absolute spectrum of a spiral wave. We then
state a convergence theorem  for spectra of linearizations along
spiral waves on large bounded disks, as the diameter of the disk tends
to infinity.

We return to the boundary-value problem~(\ref{e:evas}),(\ref{e:bc}),
with dispersion relation (\ref{e:drper3}). For $\lambda$ fixed, there
is a countable family of spatial Floquet exponents
$\nu_j=\nu_j(\lambda)$ with $0\leq\Im\nu_j(\lambda)< k_{\infty}$. The
normalization eliminates trivial shifts by $\rmi k_{\infty}$. Each of
these Floquet exponents can be seen to be of finite multiplicity as a
root of the analytic dispersion relation (\ref{e:drper3}). We order the Floquet
exponents with increasing real part, repeated by multiplicity
\begin{equation}\label{e:order}
\ldots\leq \Re\nu_{-j-1}\leq\Re\nu_{-j}\leq\ldots\leq
\Re\nu_{-1}\leq\Re\nu_{1}\leq \ldots\leq
\Re\nu_{j-1}\leq\Re\nu_{j}\leq\ldots.
\end{equation} 
Floquet exponents $\nu$ depend on $\lambda$ and the labeling might be
discontinuous in $\lambda$. We omitted the label $0$ in the counting,
for convenience. As already pointed out, the sequence is bi-infinite,
that is, $\nu_j\to\pm\infty$ for $j\to\pm\infty$. 

At this stage, we could shift all labels, $j\mapsto j + j'$ and find an equivalent
labeling. However, the Morse index considerations in the
Section~\ref{s:3.3.3} suggest a normalization at
$\Re\lambda=+\infty$. Indeed, for $\Re\lambda$ positive and large, all
$\nu_j(\lambda)$ have nonvanishing real part and we can write with a
consistent splitting
\begin{equation}\label{e:nor}
\ldots\leq 
\Re\nu_{-1}(\lambda) <  0 <\Re\nu_{1}(\lambda)\leq
\ldots.
\end{equation}
Recall that the $\nu_j(\lambda)$ change sign when $\lambda$ crosses one
of the Fredholm borders $\Gamma^\ell$. 

With normalization (\ref{e:nor}), we continue the labeling for
all values of $\lambda$. We define the {\em absolute spectrum} of
${\cal L}_*$ through
\begin{equation}\label{e:defabs}
{\rm spec}_{\rm abs}\,{\cal L}_*:=\{\lambda;\;
\Re\nu_{-1}(\lambda)=\Re\nu_{1}(\lambda)\}.
\end{equation}
As a first conclusion from the definition we note that the absolute
spectrum is a subset of the essential spectrum of ${\cal
L}_*$. Indeed, either $\Re\nu_{-1}(\lambda)=0$ and the relative Morse
index jumps at $\lambda\in{\rm spec}_{\rm abs}\,{\cal L}_*$, or else 
 $\Re\nu_{-1}(\lambda)=\Re\nu_{1}(\lambda)\not=0$ and $i_{\rm
Morse}(\lambda)\not=0$; see Proposition~\ref{p:fredisp}.

Note next that for $\lambda$ in the complement of the absolute
spectrum, ${\cal L}_*-\lambda$ is Fredholm of index zero in an
exponentially weighted space $L^2_\eta$, with radial weight
$\eta=\eta(\lambda)$ satisfying
\begin{equation}
\label{e:split}
\ldots\leq 
\Re\nu_{-1}(\lambda) <  -\eta(\lambda) <\Re\nu_{1}(\lambda)\leq
\ldots
\end{equation}
Similar to the definition in Section~\ref{s:2}, we can define the
{\em extended point spectrum} ${\rm spec}_{\rm expt}\,{\cal L}_*$ and
the {\em  boundary spectrum} ${\rm spec}_{\rm bdy}\,{\cal L}_*$.

We now characterize the limiting behavior of the spectra on
bounded disks. As in the case of Theorem \ref{t:trunc}, we omit a
number of technical assumptions in the statement of the following
theorem.
\begin{Theorem}\cite{SStrunc,SSspiralstability}
\label{t:trunc2}
Let ${\cal L}_*$denote the linearization operator (\ref{e:splin})
considered on $L^2(\R^2)$. Let ${\cal L}_{*,R}$ denote the corresponding
operator on a disk, $|x|\leq R$, equipped with typical mixed
boundary conditions. Then the spectrum of  ${\cal L}_{*,R}$ converges as
$R\to\infty$, 
\[
{\rm spec\,}{\cal L}_{*,R}\ \longrightarrow\ {\rm spec\,}_{\rm abs}({\cal
L}_*) \cup {\rm spec\,}_{\rm expt}({\cal L}_*) \cup {\rm spec\,}_{\rm
bdy}({\cal L}_*).
\] 
Convergence is uniform on bounded subsets of the complex plane in the
symmetric Hausdorff distance. Moreover, multiplicity of the
eigenvalues is preserved: the number of eigenvalues, counted with
multiplicity, in any fixed open neighborhood of any point $\lambda_*\in
{\rm spec\,}_{\rm abs}({\cal L}_*)$ converges to infinity as
$R\to\infty$. Multiplicities in neighborhoods of points $\lambda_*\in
{\rm spec\,}_{\rm expt}({\cal L}_*),\,{\rm spec\,}_{\rm pt}({\cal
L}_*),{\rm spec\,}_{\rm bdy}({\cal L}_*) $ stabilize and convergence of
eigenvalues is of exponential rate in $R$, there.
\end{Theorem}
For transverse Archimedean spiral waves, the rightmost curve
$\Gamma^0=\{\lambda^0(\rmi k)\}$ of the essential spectrum is simple,
that is, only a single Floquet exponent $\nu=\nu_{1}(\lambda)$ is
located on the imaginary axis. In particular, the {\em absolute
spectrum} does not contain $\lambda=0$. If $\Gamma^0\cap\rmi\R
=\{0\}$, and all other curves $\Gamma^\ell$ are contained in
$\Re\lambda<0$, then the absolute spectrum of the spiral wave is
stable, i.e. contained in $\Re\lambda<0$. Since the Fredholm borders
$\Gamma^\ell$ consist of the Floquet exponents of the one-dimensional
wavetrains, (\ref{e:linwt}),(\ref{e:linradinf}), stability of the
absolute spectrum is equivalent to one-dimensional stability of the
wavetrains.

This explains, on a spectral level, why spirals waves appear to be very
stable, despite the abundance of critical spectrum winding up the
imaginary axis. If the spiral is considered as a solution in a bounded
disk, only finitely many eigenvalues are close to criticality in a
bounded domain --- uniformly in the size $R$ of the domain, if the
absolute spectrum is stable.

We may say slightly more about the spectral properties of
spirals. Rotation and translation of the spiral cause $0$ and
$\pm\rmi\omega_*$ to be in the spectrum: eigenfunctions are given
explicitly by $\partial_\psi q_*(r,\psi)$, and
$\rme^{\pm\rmi\psi}(\partial_r\pm \frac{\rmi}{r}\partial_{\psi})
q_*(r,\psi)$, respectively. In particular, these eigenfunctions are
bounded functions of $(r,\psi)$. Considering the linearization in
$L^2_\eta$ with radial weight $\eta<0$ small, the linearization ${\cal
L}_*$ is Fredholm with index zero, by Definition~\ref{d:sw}, due to
the positive group velocity of the wavetrains. Because $L^2_\eta$
allows for (small) exponential growth, radially, translation and
rotation eigenfunctions contribute to the kernel. In other words,
$\lambda=0$ and $\lambda=\pm \rmi \omega_*$ belong to the extended
point spectrum and therefore, these eigenvalues stay exponentially
close to the imaginary axis when truncating to a bounded domain. 

Conversely, consider an Archimedean spirals with negative group
velocity, violating Definition~\ref{d:sw} of a {\em transverse}
Archimedean spiral. Then the essential spectrum is pushed into the
stable complex plane by an exponential weight $\exp(\eta r)$ with
$\eta>0$! As opposed to Figure~\ref{f:spiralfred}, translation and
rotation eigenvalues now do not belong to the extended point spectrum of
${\cal L}_*$, since the exponential weight requires exponential
decrease of eigenfunctions. As a consequence, Euclidean equivariance
does not imply critical spectrum of spirals with negative group
velocity. Skew-product descriptions of tip motions as described in
Section~\ref{s:meander} is not possible, and even the abstract notion
of a tip, (\ref{3.3.20}) is not well defined in this case.

\subsubsection{Point spectrum and the shape of eigenfunctions}
\label{s:3.3.4}
In the following, we will show how simple, temporal point spectrum
$\lambda_0$ of the linearization${\cal L}_*$ along an Archimedean
spiral $q_*(r,\varphi-\omega_* t)\to q_\infty(k_\infty r
-(\varphi-\omega_* t))$ gives rise to super-spiral patterns, near
resonances $\lambda_0\sim\rmi\ell\omega_*$. See expansion
(\ref{e-supspi3}), below. We prepare the resonance consideration with
a more general result, which describes the radial asymptotics of the
eigenfunction $u_0(r,\psi)$, related to any simple, temporal
eigenvalue $\lambda_0$; see Proposition~\ref{p:asyeig}.

Consider a simple eigenvalue $\lambda_0$ in the extended point
spectrum of ${\cal L}_*$, as defined in Section~\ref{s:abssp}, with
eigenfunction $u_0(r,\psi)$. Consider the roots
$\nu_j^0=\nu_j^0(\lambda_0)$ of the dispersion
relation~(\ref{e:drper3}), associated with the asymptotic wavetrains,
and the ordering (\ref{e:order})
\begin{equation}\label{e:order1}
\ldots\leq \Re\nu_{-j-1}^0\leq\Re\nu_{-j}^0\leq\ldots\leq
\Re\nu_{-1}^0<\Re\nu_{1}^0 \leq \ldots\leq
\Re\nu_{j-1}^0\leq\Re\nu_{j}^0\leq\ldots.
\end{equation} 
Note that $\Re\nu_{-1}^0<\Re\nu_{1}^0$ since $\lambda_0$ does not belong
to the absolute spectrum (\ref{e:defabs}). Assume in addition that
$\nu_1^0$ is a simple root of the dispersion relation (\ref{e:drper3}).
\begin{Proposition}\cite{SSsuperspiral}\label{p:asyeig}
In the setting of the preceding paragraph, the
asymptotic shape of $u_0$ for $r\to\infty$ is given by
\begin{equation}\label{e:asyeig}
|u_0(r,\psi)-u_\infty(r-({\psi}/{k_{\infty}}))\rme^{-\nu_{1} r}| =
\rmo(\rme^{-\nu_{1}r}),\end{equation} uniformly in $\psi$. Here
$u_\infty(\xi)$ is a spatially periodic solution to the Floquet problem
(\ref{e:evas0}) of the radially asymptotic wavetrain $q_\infty$ with
$\lambda=\lambda_0$.
\end{Proposition}
Consider nonzero $u_\infty$, in Proposition~\ref{p:asyeig}. Then
exponential radial decay or growth of $u_0$ is determined by the real
part of the first Floquet exponent $\nu_1$ of the asymptotic wavetrain
$q_\infty$. Also the temporal eigenvalue $\lambda_0$ and the radial
decay rate $\nu_1^0$ of its eigenfunction $u_0$ are related by the
dispersion relation (\ref{e:drper3}) of the wavetrain.

We now turn to the particular case of a near-resonant, simple,
imaginary eigenvalue
\[
\lambda_0=\rmi\ell\omega_* + \rmi\delta\omega,
\]
where $\delta\omega$ indicates a small deviation from the
$\ell:1$-resonance with the spiral frequency $\omega_*$. 

At resonance, $\delta\omega=0$, we obtain explicitly
\begin{equation}\label{e:asyres}
u_\infty(r-\psi/k_\infty)=\rme^{-\rmi\ell\psi}q'_\infty(r-\psi/k_\infty),
\end{equation} 
simply because $q_\infty'(\xi)$ is the Floquet solution of the trivial
Floquet exponent $\nu_1=0$ and due to the Floquet shift
(\ref{e:flsp}). 

Slightly off resonance, that is, for small nonzero $\delta\omega$, we
observe associated perturbations $\delta u_\infty$, of the radially
asymptotic Floquet eigenfunction $u_\infty(\xi)$, and $\delta\nu_1$,
of the corresponding Floquet exponent. In view of
Proposition~\ref{p:asyeig}, $\delta u_\infty$ remains $2\pi$-periodic,
as $u_\infty$ itself, and has little effect on the observed pattern. A
variation of $\Re\delta\nu_1\not=0$ modifies the radial exponential
decay or growth of the eigenfunction $u_0$. From the discussion in
Section~\ref{s:3.3.3} on crossing of Floquet exponents near Fredholm
borders, Proposition~\ref{p:fredisp}, we predict exponential growth in
the Fredholm index -1 region and exponential decay in the Fredholm
index 0 region; see Figure~\ref{f:spiralfred}. However,
$\Re\delta\nu_1\not=0$ does not influence the $r$-asymptotic shape of
the pattern. Only $\Im\delta\nu_1\not=0$ interferes with the
underlying periodicity of the primary spiral pattern, producing
super-spiral patterns; see Figure \ref{f:meanderpics}. More
specifically,
\begin{eqnarray}
\label{e-supspi}
u_0(r,\psi) & \sim & \rme^{-\nu_1^0 r} u_\infty(r-\psi/k_\infty) =
\rme^{-\delta\nu_1 r} \rme^{-\rmi\ell\psi} \tilde{q}_\infty'(\xi) \\ & = &
\rme^{-(\Re\delta\nu_1) r} \rme^{-\rmi\ell\psi}\tilde{q}_\infty'(\xi)
\rme^{-\rmi(\Im\delta\nu_1) r}.\nonumber
\end{eqnarray}
Here, $\tilde{q}_\infty'(\xi)$ is a small, but $2\pi$-periodic
perturbation of the wavetrain eigenfunction ${q}_\infty'(\xi)$, and
$\sim$ indicates identity up to a term of order $\rmo(\rme^{-\nu_1^0
r})$, for $r\to\infty$. The term $\rme^{-\rmi\ell\psi}$ again accounts
for the Floquet shift of $u_\infty$.  Since $\nu_1=0+\delta\nu_1$
satisfies the dispersion relation (\ref{e:drper3}), the definition of
the group velocity $c_{\rm g}$ (\ref{e:cg3}) implies
\begin{equation}\label{e-supspi2}
\Im\delta\nu_1 = \frac{\Im\delta\nu_1}{\Im\delta\lambda_1} \cdot
\Im\delta\lambda_1 = \frac{\delta\omega_*}{c_{\rm g}},
\end{equation}
in the resonant limit $\delta\omega \to 0$. Substituting
(\ref{e-supspi2}) into (\ref{e-supspi}), and returning to the original
coordinates $r,\varphi=\psi+\omega_* t$, and time $t$, the
eigenfunction generates a perturbation 
\begin{equation}\label{e-supspi3}
u_{\rm pert}(t,r,\varphi)\sim\rme^{-(\Re\delta\nu_1)r}\cdot
\rme^{\rmi(\ell\varphi - (r-c_{\rm g} t)\delta\omega/c_{\rm g})}\cdot
\tilde{q}_\infty'(r-(\varphi-\omega_* t)/k_\infty),
\end{equation}
again for $r\to\infty$, and in the limit $\delta\omega\to 0$. 

The eigenfunction shape $u_{\rm pert}$, corresponding to a
near-resonant temporal eigenvalue $\lambda_0=\rmi\ell\omega_* +
\rmi\delta\omega$, produces the pattern of a superspiral, for large
$r$. Indeed, consider fixed $t$. Then $u_{\rm pert}$ is an
$\ell$-armed spiral, due to the term
$\exp(\rmi\ell\varphi-\delta\omega r/c_{\rm g})$. The superspiral is
Archimedean with large radial wavelength $2\pi c_{\rm
g}/\delta\omega$, for $\delta\omega\to 0$. The superspirals propagate
in radial direction with speed $c_{\rm g}$ and rotate with the slow
frequency $\delta\omega$. 

In the experimentally observed pattern $q_*+\varepsilon u_{\rm pert}$,
the superspiral modulations manifest themselves in a phase shift of
the emitted wavetrains
\begin{equation}\label{e:supspi4}
q_*+\varepsilon u_{\rm pert}\sim q_\infty( r - (\varphi-\omega_*
t)/k_\infty + \rme^{-(\Re\delta\nu_1)r}
\cos(\ell\varphi - (r-c_{\rm g} t)\delta\omega/c_{\rm g})).
\end{equation}
Here we exploited that $\tilde{q}_\infty'\sim {q}_\infty'$ in
(\ref{e-supspi3}) amounts to a phase shift of the emitted wavetrains.

\subsection{Comparison with experiments}
\label{s:4.n}
We explain the onset of spatio-temporal patterns and certain characteristics of incipient  
spiral wave instabilities, referring to the linear spectral analysis of the
preceding section. In particular, we discuss the possibility of point
spectrum crossing the imaginary axis, Section~\ref{s:3.4.2}, and
essential spectrum causing an instability, Section~\ref{s:3.4.3}.

\subsubsection{Meander instabilities}
\label{s:3.4.2}
Meander instabilities are caused by critical point spectrum, emerging
from the essential spectrum; see Figure~\ref{f:meanderspec}. In any
disk of finite radius $R$, the essential spectrum disappears and five
critical eigenvalues remain close to the imaginary axis. The influence
of the boundary of the domain on the location of point spectrum  is
exponentially small for large disk radius $R$. Actually, also the
nonlinear terms of the vector field on the center-manifold converge
with exponential rate, as the size $R$ of the disk tends to
infinity ~\cite{SSmea}. In the limit, we recover the $SE(2)$-equivariant
skew-product description (\ref{3.3.17}). The equations on the center
manifold in a finite disk are therefore a small perturbation of the
$SE(2)$-invariant skew-product system, discussed in
Section~\ref{s:meander} --- as long as the tip stays far enough from
the disk boundary.

Therefore, in any large disk, we recover the skew-product dynamics,
even though a dynamical reduction in the unbounded domain is
prohibited by the presence of essential spectrum.

\begin{figure}[h]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,6.)%
\centerline{\includegraphics[scale=0.4]{Fig/meander-spec.eps}}
\end{picture}%
\caption[]{Schematic plot of the meander eigenvalues popping out of
the Fredholm index~-1 region close to instability.}  
\label{f:meanderspec} 
\end{figure}
Dynamics and tip motion are complemented in the experiments with
striking super-patterns in the farfield, when sufficiently large
spirals are observed. As a consequence of center manifold reduction,
the spatial patterns are superpositions of the critical
eigenfunctions, to leading order in the bifurcation parameter. For two
purely imaginary eigenvalues close to resonance $\rmi\omega_{\rm
H}\sim\ell \rmi\omega_*$ we have computed the shape;
see~(\ref{e:supspi4}). Predictions coincide well with direct
simulations of a specific reaction-diffusion system, based on
Barkley's code EZSpiral~\cite{domaba97}; see
Figure~\ref{f:meanderpics}. For marginally stable wavetrains, the
essential spectrum touches the imaginary axis only at $\rmi\ell
\omega_*$, $\ell\in\Z$, with quadratic tangency. From the previous
section, we conclude that, off resonance, super-patterns $u_{\rm pert}$
resulting from the instability will be strongly localized in the
farfield~(\ref{e-supspi3}). In particular, at the higher {\em
drift}-resonances, $\omega_*=\ell\omega_{\rm H}$, $\ell\geq 2$, the
super-patterns will be strongly localized. At the inverse resonances,
$\ell\omega_*=\omega_{\rm H}$, $\ell\geq 2$ where the motion of the
tip stays bounded, we do predict prominent, non-localized
super-patterns in the form of weakly curved $\ell$-armed
super-spirals~(\ref{e-supspi3}). The orientation of the super-spirals
changes when the Hopf eigenvalue crosses through resonance. At
transition, the super-pattern divides the domain in sectors. At the
1:1-resonance, a particularity occurs, since $\rmi\omega_*$ is now
algebraically double in the extended point spectrum. Drift resonance
coincides with the emergence of strongly pronounced
super-spirals. In~\cite{SSsuperspiral}, it is shown that the principal
eigenfunction is computed from integration of the eigenfunction in the
radial direction. Its amplitude grows linearly with the radius $r$,
such that the super-pattern exhibits an asymptotically $r$-independent
shift of the wavenumber of the wavetrains; see the drift picture in
Figure~\ref{f:meanderpics}.

Experimentally, these striking superstructures, have first been
observed in a biological system~\cite{PMExp}. An explanation based on
a curvature description of the emitted pulse-trains, in the spirit of
Section~\ref{s:kin}, has been attempted in~\cite{PMtheory}. The simplest
phenomenological explanation evokes the picture of a point source,
alias the tip, emitting wavetrains. Variations of the local
wavenumber are due to a Doppler-effect caused by the two-frequency
motion of the spiral tip. Spatial decay as well as the correct speed
of propagation are not predicted from these formal descriptions, which
neglect the interaction between individual emitted pulses.

\begin{figure}[h]
\setlength{\unitlength}{1cm}%

\hfill\includegraphics[scale=0.125]{Pics/m1.eps}%
\hfill\includegraphics[scale=0.125]{Pics/m2.eps}%
\hfill\includegraphics[scale=0.125]{Pics/m3.eps}\hspace*{\fill}

\vspace{-1.31cm}
\hspace*{3.5cm}\includegraphics[scale=1.0]{Pics/m1s.eps}%
\hspace{3.5cm}\includegraphics[scale=1.0]{Pics/m2s.eps}%
\hspace{3.5cm}\includegraphics[scale=1.0]{Pics/m3s.eps}

\caption[]{Farfield  patterns and tip motion in case of inward
meander, drift, and outward meander. The large pictures show the
superspiral-shaped deformations of the primary spiral. The small
inlets show, on a smaller spatial scale, the tip motion.}  
\label{f:meanderpics} 
\end{figure}

The spectral assumptions of point spectrum crossing the axis were
confirmed numerically by Barkley; see~\cite{Barkley92}. In a disk, he
found five critical eigenvalues at bifurcation. The eigenvalues from
translation were extremely close to the axis, although the domain size
was moderate, exhibiting about one turn of the spiral. This confirms
the rapid (exponential) convergence of point spectrum in bounded
domains of increasing size $R$; see Theorem~\ref{t:trunc}. The
numerically observed change of the growth behavior of the amplitude of
the eigenfunction is in good agreement with the predictions from
Section~\ref{s:3.3.4}. Indeed, close to resonance, (\ref{e-supspi3})
predicts that the asymptotic behavior close to bifurcation changes:
from exponential radial growth --- when the eigenvalue is still
located in the Fredholm index -1 region --- to weak exponential radial
decay at bifurcation.  


\subsubsection{Farfield and core breakup}
\label{s:3.4.3}
Farfield breakup is caused by essential spectrum crossing the
imaginary axis~\cite{BO,SSspiralstability,TK}. Mostly, a convective
Eckhaus instability is observed, with unstable wavenumbers close to
the wavenumber of the wavetrains, $\Im\nu\sim k_{\infty}$. In the
stable regime, the boundary curve $\Gamma$ of the essential spectrum
touches the imaginary axis at $\rmi\omega_*\Z$ with quadratic tangency
from the left. At criticality, the tangency is of fourth order. In the
unstable regime, the tangency is from the right; see
Figure~\ref{f:spectrafarfield}. The absolute spectrum, however,
remains confined in the open left half plane, close to this
instability threshold. The group velocities (\ref{e:cg3}) on the
critical spectral curve $\Gamma$ are always directed outwards close to
$\lambda=0$, by transversality of the spiral; see
Definition~\ref{d:sw}. Therefore, the Eckhaus unstable modes inherit a
positive group velocity, as well. In any large bounded disk, the
Eckhaus instability remains transient, as long as the absolute
spectrum stays confined to the left complex half plane. This was
confirmed in direct simulations, see~\cite{SSspiralstability}. We
computed essential spectra of the wavetrains from the boundary-value
problem~(\ref{e:evas}),(\ref{e:bc}) following the dependence of a
nontrivial solution $u(\xi; k),\lambda(\rmi k)$ on the Bloch
wavenumber $ k$. The absolute spectrum was computed by a similar
path-following procedure. Instability thresholds coincided well with
the observations in direct simulations. In the convectively unstable
regime, stability depends crucially on the size of the domain and the
magnitude of the perturbation. The sensitivity is caused by the
subcritical nature of the Eckhaus instability in the given parameter
regime. Qualitatively, the sensitivity mimics the one-dimensional
nonlinear advection-diffusion problem, addressed in
Proposition~\ref{p:nonlin}: the minimal amount $\delta^{\rm u}$ of a
perturbation needed to cause breakup decays rapidly with the size $R$
of the domain. In numerical simulations, again based on Barkley's
code EZSpiral~\cite{domaba97}, the domain contained approximately 20
wavelengths; see Figure~\ref{f:breakup-superspirals}.

The absolute instability, if close enough to the convective
instability, is produced by eigenfunctions which grow exponentially
towards the boundary of the domain; see
Figure~\ref{f:spectrafarfield}. Their shape is again determined from
the complex dispersion relation (\ref{e:drper3}), which predicts
prominent, exponentially growing super-spirals (\ref{e-supspi3}). Note
that these super-spirals resemble the farfield pattern in the meander
instability. The most striking difference is their growth towards the
boundary. However, we emphasize that super-spirals in meandering and
in farfield breakup occur for completely different reasons. In the
farfield breakup, the motion of the tip is a regular one-frequency
motion.

\begin{figure}[h] 
\setlength{\unitlength}{1cm}% 
\begin{picture}(13.5,4.9)% 
\centerline{\includegraphics[scale=0.4]{Fig/spec-farfield.eps}}
\centerline{\includegraphics[scale=0.4]{Fig/spec-farfield-abs.eps}}
\end{picture}% 
\caption[]{Schematic plot of essential and absolute spectra crossing
the axis successively at farfield breakup.}
\label{f:spectrafarfield}  
\end{figure} 

We discuss core breakup, next. Numerical computations of essential and
absolute spectra indicate that parts of the critical curve $\Gamma$
change slope to create group velocities directed towards the center of
the spiral. The shape of the boundary curve of the essential spectrum
becomes more complicated. In Figure~\ref{f:spectracore}, two scenarios
are suggested, that produce negative group velocities. Recall that the
sign of the group velocity determined the vertical slope of the
oriented curve $\Gamma^\ell$. The onset of the second scenario in
Figure~\ref{f:spectracore} has been found in~\cite{SSspiralstability}
in a parameter regime close to core-breakup. In direct simulations of
a nonlinear reaction-diffusion system, super-spirals appear before
breakup, but are strongly localized. Much more detailed numerical
investigation is necessary, however, to clarify the structure of this
instability.
\begin{figure}[h]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,3.7)%
\includegraphics[scale=1.0]{Fig/mushrooms.eps}
\end{picture}% 
\caption[]{Two scenarios (a),(b) of  essential and absolute spectra crossing
the axis successively at core-breakup.}
\label{f:spectracore} 
\end{figure}

\begin{figure}[h]
\setlength{\unitlength}{1cm}%

\hfill\includegraphics[scale=0.175]{Pics/c1.eps}%
\hfill\includegraphics[scale=0.175]{Pics/f1.eps}%
\hspace*{\fill}

\caption[]{Farfield  patterns in case of farfield- and core-breakup.}
\label{f:breakup-superspirals} 
\end{figure}

The final states in farfield versus core breakup are quite
different. Collision of wavetrains leads to the formation of new
spirals. In the case of farfield breakup, the size of the various
spiral domains remains large, close to the onset of instability,
leading to a moderately complicated dynamical behavior. In the
opposite case of core breakup, new spirals immediately break up, close
to the core, and the final state is extremely complicated right after
the instability has set in; see Figure~\ref{f:finalstate}.

\begin{figure}[h]
\setlength{\unitlength}{1cm}%

\hfill\includegraphics[scale=0.175]{Pics/f3.eps}%
\hfill\includegraphics[scale=0.175]{Pics/c3.eps}%
\hspace*{\fill}

\caption[]{Final state after farfield and core breakup.}
\label{f:finalstate} 
\end{figure}

Meander instabilities in the presence of convective farfield
instabilities enhance the effect of the instability. From the spectral
analysis in Section~\ref{s:3.3} we predict the meander eigenfunctions
to be exponentially growing. Indeed, at instability, they are embedded
in the Fredholm index -1 region.  Whereas the unstable essential
spectrum disappears in bounded domains, the (extended) point spectrum
still reproduces the effect of the essential spectrum. Indeed 
exponential radial growth of eigenfunctions is governed by the
dispersion relation~(\ref{e:drper3}).  In the simplistic phenomenological
description by a Doppler effect, the meander movement of the spiral
tip is amplified and transported by the wavetrains. Increasing the
size of the domain, we find breakup in the farfield already for
moderate amplitudes of the meander motion.




\section{Three space dimensions: scroll waves}
\label{c-4}
\subsection{Filaments, scrolls, and twists}
\label{s-4.1}
In this chapter we consider reaction-diffusion systems of $N=2$
equations in $m=3$ space dimensions,
% 
\begin{eqnarray}u_t = D\Delta u &+& f(t,x,u). \label{4.1.1}\end{eqnarray}
Here $u=(u_1,u_2)\in\mathr^2$ and $x$ is in a bounded domain
$\Omega\subset \mathr^3,$ say with Neumann boundary conditions. Let
$f$ be smooth. A motivating prototype is the singularly perturbed
FitzHugh-Nagumo-like system with nonlinearity
% 
\begin{eqnarray}
f &=& \left( \begin{array}{c} \varepsilon^{-1}u_1
(1-u_1)(u_1-(u_2+b)/a) \\ 
u_1-u_2 \end{array} \right) 
\label{4.1.2}\end{eqnarray} 
of excitable media type, for small $\varepsilon >0.$ See
Section~\ref{s:kin}, and (\ref{3.1.18}). For simplicity of notation,
we shift $u=(u_1,u_2)$ by suitable constants, in each component, so
that the value $u=0$ becomes a value inside the singular homoclinic
pulse cycle of Figure~\ref{Fig3.3}(a). 

\subsubsection{Spirals, tips, and Brouwer degree}
In $m=2$ space dimensions,
$\Omega\subset\mathr^2,$ the pre-images
% 
\begin{eqnarray}z_*(t) =(u(t,\cdot))^{-1}(0) \label{4.1.3}\end{eqnarray}
may serve as a viable definition for a spiral tip in the sense of
(\ref{3.3.20}), (\ref{3.3.21}). Both experimental evidence and
numerical simulations suggest, in fact, that nondegenerate zeros $z_*$
of $u(t,\cdot)$ are typically accompanied by local spiral wave
patterns of (\ref{4.1.2}) with rotation direction determined by the
local Brouwer degree (see for example
\cite{deimli85,dufono85,zeidle93} for some background):
% 
\begin{eqnarray}
\deg_{loc}u(t,\cdot) &:=& {\rm sign\;} \det u_x(t,z_*(t)) 
\label{4.1.4}\end{eqnarray}
at $x=z_*(t).$ So pronounced is this effect, indeed, that Winfree has
proposed to model excitable media by only a phase variable
$\tilde{u}=\tilde{u}(t,x)\in S^1,$ with appropriate time evolution
\cite{winfre80,winfre87,winfre95}. Here the circle $S^1$ can model the
homoclinic pulse cycle of Figure~\ref{Fig3.3}(a), or else just an
abstract local ``clock''. Mathematically, the correspondence to our
PDE approach can be achieved by simply defining $\tilde{u}:=u/|u|.$
The local Brouwer degree (\ref{4.1.4}) coincides with the winding
number of $\tilde{u},$ restricted to a small circle around the
singularity of $\tilde{u}$ at $x=z_*$. Sometimes this local degree is called
``topological charge'' of the singularities of spirals. Conservation
of ``charge'', as time $t$ evolves, then corresponds to homotopy
invariance of Brouwer degree: only anti-rotating spiral pairs can
annihilate each other.

The restriction to $\tilde{u}\in S^1$ unfortunately produces a phase
singularity of undefined $\tilde{u}$ at the spiral tips, where $u=0.$
Since we are moreover interested in collisions of such singularities,
not only in two but also in three space dimensions, we therefore favor
the reaction-diffusion description by smooth spatial profiles
$u(t,\cdot)\in\mathr^2$ over the phase fields $\tilde{u}(t,\cdot)\in
S^1.$

\subsubsection{Scroll waves, filaments, and twists}

Returning to three-dimensional $x\in\Omega\subset\mathr^3,$ the preimages
% 
\begin{eqnarray}\tau \mapsto z=z(t,\tau) \in
(u(t,\cdot))^{-1}(0)\subseteq \Omega \label{4.1.5}\end{eqnarray} 
become differentiable curves $z(t,\cdot),$ at least as long as zero is
a regular value of $u(t,\cdot).$ We call these curves {\em (tip)
filaments.} Numerical observations in excitable media exhibit spiral
wave patterns, when restricting the spatial profile $u(t,\cdot)$ to a
local, two-dimensional section $S^\bot\subset\Omega$ transverse to the
filament curve $z(t,\cdot)$ at $z_*=z(t,\tau_0)$. Note, that we may
always define an orientation of $S^\perp$ such that
% 
\begin{eqnarray}\deg_{loc} u(t,\cdot) |_{S^\perp} = +1
\label{4.1.6}\end{eqnarray} 
at $x=z_*.$ Requesting the pair $(S^\perp,z_\tau)$ of $S^\perp$ and
the filament tangent $z_\tau$ to be oriented by the right-hand rule
thus defines a unique {\em filament orientation} all along each
filament curve $\tau\mapsto z(t,\tau).$

A {\em scroll wave} is the wave pattern arising in a tubular
neighborhood of a filament curve.  By the excitable media paradigm,
the scroll wave consists of a stack of spirals in the transverse
sections $S$ along the filament. See Figure \ref{Fig4.1} for a
reproduction of Panfilov and Winfree's original drawings~\cite{panwin85}. We
emphasize that our mathematical discussion of scroll waves neither
proves nor relies on the validity of the ``stack of spirals''
picture. Our results below will hold in a general framework of
reaction-diffusion systems. In particular, they are not limited to a
singular perturbation setting. It is nevertheless very useful to refer
to Winfree's paradigm, for visual motivation and guidance.

For closed filaments $z(t,\tau),\tau\in S^1,$ a
{\em twist} of the oriented filament can be defined as follows. Choose
$\delta >0$ small enough and consider the closed curve
% 
\begin{eqnarray}
\tau\mapsto\tilde{z}(t,\tau) \subseteq (u(t,\cdot))^{-1}(\delta,0)
\label{4.1.7}\end{eqnarray} 
of the pre-image of $u_1=\delta,u_2=0$ near $z(t,\tau),$ following the
orientation of $z$ itself. Geometrically, the curve
$\tilde{z}(t,\cdot)$ indicates the local phases of the spiral motions
in the stacked sections $S^\perp.$ Then we call
% 
\begin{eqnarray}
{\rm twist}(z(t,\cdot)) & :=& {\rm lk}(\tilde{z}(t,\cdot), z(t,\cdot))
\label{4.1.8}\end{eqnarray} 
the twist of the filament $z(t,\cdot).$ Here lk denotes the {\em
linking number} of oriented curves, as explained below. A {\em twisted
scroll} possesses a filament $z$ with nonzero twist: the curves
$z(t,\cdot)$ and $\tilde{z}(t,\cdot)$ are linked.

%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
%\begin{picture}(13.5,10.3)%
\centerline{\includegraphics[scale=1.0]{Fig/Fig27a.eps}}
%\end{picture}%
\vspace*{1cm} %1cm
\caption{Scroll waves are stacks of spirals (a). Twisted scroll wave (b). Courtesy of~\cite{panwin85}.}  \label{Fig4.1}
\end{figure}


To define the {\em linking number} ${\rm lk\,}(\tilde{z},z)$ of two closed
oriented disjoint curves
$\tilde{z}=\tilde{z}(\tau),z=z(\tau),\;\tau\in S^1,$ we consider a
suitable planar projection such that crossings of the two curves are
transverse and distinct, in projection. See Figure \ref{Fig4.1.2}, for
examples of such projected crossings, indicating over- and
under-crossings in the original three-dimensional view. Denoting the
oriented unit tangent vectors of the over- and under-crossing curves by
${\bf t}_1,{\bf t}_2,$ respectively, we associate a sign
% 
\begin{eqnarray}\varepsilon_j & :=& {\rm sign}\det ({\bf t}_1,{\bf t}_2)=\pm 1 \label{4.1.8b}\end{eqnarray} 
%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,3.3)%
\centerline{\includegraphics[scale=0.8]{Fig/Fig25.eps}}
\end{picture}%
\caption{Transverse crossings of planar projections of oriented
curves, with associated signs $\varepsilon_j=\pm 1.$ Note ${\rm
lk}\,(z,\tilde{z})=+1$, and ${\rm twist}\,(z)={\rm twist}\,(\tilde{z})=-1$.}
\label{Fig4.1.2}
\end{figure}
to each crossing $j$. Then
% 
\begin{eqnarray}{\rm lk}\,(\tilde{z},z) & :=& \frac 1 2 \sum_j\varepsilon_j, \label{4.1.8c}\end{eqnarray} 
where the sum extends over all mutual crossings, but not over
self-crossings. Clearly, ${\rm lk}\, (\tilde{z},z)={\rm lk}\,(z,\tilde{z}).$
The linking number ${\rm lk}\,(\tilde{z},z),$ and hence ${\rm twist}\,(z)$, is
independent of the choice of small enough $\delta >0.$ The linking
number does not depend on the choice of projection and is invariant
under diffeotopies of $\mathr^3;$ see for example \cite{kauffm87}. In
particular, ${\rm lk}\,\neq 0$ implies ${\tilde z},z$ cannot be unlinked,
for example by a diffeotopy such that ${\tilde z},z$ come to lie in
the same plane.

The following proposition indicates how twisted scrolls have to be
either knotted or else have to be linked to other twisted scrolls.

%
\begin{Proposition}\label{p-4.1}
Let $S\subset \Omega$ be an embedded surface, diffeomorphic to a
closed disk, such that $\partial S$ is a closed filament curve
$z(t,\cdot).$ Assume that $S$ is transverse to all other filament
curves $z_j(t,\cdot)\subset u(t,\cdot)^{-1}(0)$ intersecting $S$. Let
$\zeta_k$ enumerate these intersection points. Choose an orientation
for $S$ such that the induced boundary orientation of $\partial S$
coincides with the orientation of the filament $z(t,\cdot).$ Then
\begin{eqnarray}
{\rm twist\;}(z) & + & \sum_{x=\zeta_k} \deg_{loc}u(t,\cdot)|_S=0 
\label{4.1.9}\end{eqnarray}
\end{Proposition}
For the proof, see \cite[\S 15.4]{dufono85}.

Of course Proposition \ref{p-4.1} holds only as long as $u=0$ is a regular
value of the spatial solution profile $u(t,\cdot),$ for fixed time
$t>0$. Neither linking nor knotting can change, in this situation. In
the next section we will address the precise geometry of the failure
of this assumption, which occurs at only discrete times $t>0$, for
generic initial conditions $u_0$ at $t=0.$

\subsection{Generic changes of scroll filament topology}
\label{s-4.2}
How can two scroll wave filaments collide and ``cross'' each other as time
$t$ evolves? In Figure \ref{Fig4.2} we show two linked closed scroll
wave filaments $z_1(t,\cdot), z_2(t,\cdot)$, each parameterized over
$\tau\in S^1.$ In scenario (a), called ``{\em break through}'', the
two circular filaments interpenetrate each other and restitute
themselves as two unlinked circles $\tilde{z}_1, \tilde{z}_2$ after
collision. The ``{\em crossover}'' scenario (b), in contrast, calls
for a ``kissing'' of the two filaments $z_1(t,\cdot),z_2(t,\cdot)$ to
occur, at time $t=t_0$, with a subsequent reconnection, such that a
single closed filament $z$ is generated. Now, which scenario is right?

%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,3)%
%\centerline{
\hspace{4.3cm}\includegraphics[scale=0.6]{Fig/Fig26.eps}
\end{picture}%
\caption{Two scenarios for topology changes of linked scroll filaments: (a) break through; (b) crossover.} 
\label{Fig4.2}
\end{figure}


Note how filament orientation, as defined locally in (\ref{4.1.6}),
has to be respected by either scenario --- and it is. 
Applying the scenario of Figure \ref{Fig4.2}(a) to one of the mutual
crossings in Figure \ref{Fig4.1.2} unlinks the two closed twisted filaments $z$
and $\tilde{z}$ . Scenario \ref{Fig4.2}(b), in contrast, merges the two
filaments to a single closed, untwisted filament, when applied between
the top and bottom crossing. 


\subsubsection{Generic level sets}

To decide on the appropriate scenario we follow the approach in
\cite{fieman00}. In Section \ref{s-4.1} we have seen how the
phenomenon of scroll wave filaments $x=z(t,\tau)\in\Omega$
mathematically corresponds to zeros $(t,x)\in(0,\infty)\times\Omega$
of solutions $u=u(t,x)\in\mathr^2$ of a reaction-diffusion system:
%
\begin{eqnarray}u(t,x) &=& 0. \label{4.2.1}\end{eqnarray}
If $u=u(t,x)$ was just ``any'' function, rather than a solution to a
PDE, then singularity theory would provide ample information on the
behavior of the zero set $\{(t,x);\;u(t,x)=0\}.$ Indeed, generic
functions $u$ only possess a finite number of singularities under our
dimensional constraints. See for example \cite{arguva85,golsch85,arnold93,arnold94b} for
more details. We give specific examples below. However, $u=u(t,x)$ is
not just ``any'' function: it is a solution $u=u(t,x;u_0)$ of a
reaction-diffusion system (\ref{4.1.1}) subject to boundary conditions
and, more importantly, with prescribed initial conditions
%
\begin{eqnarray}u(t=0,x) &=& u_0(x). \label{4.2.2}\end{eqnarray}
%
\begin{Theorem}\label{t-4.2}
\cite{fieman00} Let $k\geq 1.$ Consider initial conditions $u_0$ in a
Sobolev space $X$ which embeds into $C^k(\Omega,\mathr^N)$ and
satisfies the (Neumann) boundary conditions. Then the solution $u(t,x)$
of (\ref{4.1.1}) possesses only generic singularities, as determined
by derivatives in $(t,x)$ up to order $k$, for generic initial
conditions $u_0\in X$ and at any $t>0$, $x\in\Omega\subset\R^m$.
\end{Theorem}

Note that we use the word singularity in the sense of singularity
theory here. We do not discuss ``singularities'' in the PDE-sense of
blow-up of solutions or their derivatives.
Nor are "singularities" understood as phase singularities, where some
$\tilde{u}=u/\mid u\mid \in S^1$ remains undefined. Rather, a {\em singularity } is given by an algebraic variety $S$ in
the jet space $J^k$ of Taylor polynomials up to order $k$ in the
variables $(t,x)$. Consider the map
%
\begin{eqnarray}\begin{array}{rcl} j^k:(0,\infty)\times\Omega & \rightarrow & J^k\\
(t,x) &\mapsto & j^k u(t,x) \end{array} \label{4.2.3}\end{eqnarray}
which maps $(t,x)$ to the $k$-jet $j^ku(t,x),$ the Taylor polynomial
of order $k$ of the solution $u$, evaluated at $(t,x).$ Since $t>0$
and $x\in \Omega\subset\mathr^m$, together, are only $m+1$ independent
variables, a generic function $u(t,x)$ will certainly miss a variety
$S$ unless
%
\begin{eqnarray}{\rm codim}\ S & \leq & m+1 \label{4.2.4}\end{eqnarray}
in $J^k$. The theorem states that, likewise, solutions $u=u(t,x;u_0)$
of (\ref{4.1.1}) will miss such singularities $S$, at least for a
subset $X'$ of initial conditions $u_0\in X$ which is residual in the
sense of Baire: $X'$ contains at least a countable intersection of
open dense subsets of $X.$

The proof of Theorem \ref{t-4.2} is based on Thom transversality and
backwards uniqueness for linear, nonautonomous parabolic
systems. Essentially, Thom transversality reduces the problem to
showing surjectivity of the linearization, with respect to $u_0,$ of
the $k$-jet of the solution $u$ at any fixed $(t,x)$. This
finite-dimensional surjectivity property is then proved by a backwards
uniqueness result which prevents solutions of linear equations to
become zero in finite time. For a technically more complete account
and slightly more general results we refer to \cite{fieman00}. For earlier
results in a similar direction, of a purely local singularity theory
flavor, see also \cite{damon97}.

\subsubsection{Sturm property, revisited}
We give some examples for our result, before returning to the problem
of filament crossing.  Since we are interested in zeros of solutions
$u(t,x)$, the requirement $u=0\in\mathr^N$ will always contribute $N$
to ${\rm codim} S$ in (\ref{4.2.4}). Consider scalar equations, $N=1,$
in one space dimension, $m=1$ first. To exclude $x$-profiles $x\mapsto
u(t,x)$ with zeros of multiplicity three or higher, define $S$ by the
requirement
%
\begin{eqnarray}u=u_x = u_{xx}=0. \label{4.2.5}\end{eqnarray}
Since this implies ${\rm codim} S=3>2=m+1,$ the realization condition
(\ref{4.2.4}) is violated. Therefore (\ref{4.2.5}) cannot occur, for
generic initial conditions $u_0\in X \hookrightarrow C^2.$ Similarly,
$u_t\neq 0$ at double zeros $u=u_x=0,$ because
%
\begin{eqnarray}u=u_x = u_t=0 \label{4.2.6}\end{eqnarray}
also implies ${\rm codim} S=3.$ Therefore the only remaining local singularity basically takes
the form 
%
\begin{eqnarray}u(t,x)=\pm (t-t_0)\pm (x-x_0)^2; \label{4.2.7}\end{eqnarray}
see Figure \ref{Fig4.3}. In Section \ref{s-2.1} we have seen how
$f=f(x,u,u_x)=0$ for $u=u_x=0$ implies that the two signs in
(\ref{4.2.7}) have to coincide. This Sturm property excludes case (b)
of creation of a new pair of zeros, admitting only annihilation
(a). Of course the Sturm property of Proposition~\ref{p-2.1}, unlike
our present genericity argument, is not limited to generic initial
conditions $u_0\in X.$
%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,2.5)%
\centerline{\includegraphics[scale=1.0]{Fig/Fig27.eps}}
\end{picture}%
\caption{Collision of zeros $u(t,x)=0$ for scalar equations: (a)
annihilation, (b) creation.}
\label{Fig4.3}
\end{figure}
\subsubsection{Comparison principle and nodal domains}
For scalar equations, $N=1,$ in arbitrary space dimension $m\geq 1,$
Fig. \ref{Fig4.3} remains valid. Indeed the $m+1$ conditions
%
\begin{eqnarray}u=0, &\;& u_{x_1}=\cdots =u_{x_m}=0 \label{4.2.8}\end{eqnarray}
prevent further degeneracies to appear in generic singularities $S$, according to realization
condition (\ref{4.2.4}). In particular $u_t\neq 0$, and the Hessian matrix $u_{xx}$ has to be
nondegenerate.

If the Hessian matrix $u_{xx}$ is strictly (positive or negative)
definite, spherical bubbles of dimension $m-1$ are thus annihilated
(a) or created (b). For $f=f(x,u,\nabla u)$ with $f(x,0,0)=0,$
however, creation (b) is again excluded by the partial differential
equation which involves $\Delta u= {\rm trace\;} (u_{xx}).$ In the
general, nongeneric case, this is due to the maximum (or comparison)
principle; see Section~\ref{s-2.1}, (\ref{2.1.10})--(\ref{2.1.12}).

Suppose next that the Hessian $u_{xx}$ is indefinite. For simplicity
we consider $m=\dim x=2.$ In suitable local coordinates $x=(x_1,x_2)$,
and normalizing to $x_0=0\in\Omega,$ we obtain the local form
%
\begin{eqnarray}u(t,x) &=& (t-t_0)+x_1^2-x^2_2 \label{4.2.9}\end{eqnarray}
%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,4.0)%
\centerline{\includegraphics[scale=1.0]{Fig/Fig28.eps}}
\end{picture}%
\caption{Zero sets of $u(t,x)$ according to (\ref{4.2.9}) as $t$
passes through $t_0.$} \label{Fig4.4}
\end{figure}

%
\noindent See Figure \ref{Fig4.4} for the curves describing the local zero
set. This ``Morse scenario'' of course coincides with the behavior of
level curves near a nondegenerate saddle point, with level values
parameterized by time. Indeed $N=\dim u=1$, and $u_t\neq 0$
locally. The curves are oriented such that $u<0$ to the left of each
curve. This orientation is the analogue, for $N=1$ and $m=2,$ of the
orientation of filament curves which was defined for $N=2$ and $m=3$
in Section \ref{s-4.1}. Note how the connectivity of the nodal domains
$\{u(t,\cdot) >0\},\; \{u(t,\cdot)<0\}$ changes, as $t$ increases
through $t_0.$ Because the sign of $u_t=\Delta u= {\rm trace\;}
(u_{xx})$ at $(t_0,x_0)$ is indeterminate for indefinite Hessians
$u_{xx},$ even when $f=f(x,0,0)=0$, there is no analogue to the Sturm
property in higher space dimensions $m\geq 2$ which could be based on
counting nodal domains. See however \cite{uhlenb76} for generic
simplicity and sign change properties of eigenfunctions of the
Laplacian in higher space dimensions.

\subsubsection{Annihilation of spiral tips}
Finally we return to systems, $N=\dim u=2,$ for example of the
singularly perturbed FitzHugh-Nagumo type (\ref{4.1.1}),
(\ref{4.1.2}). Consider $S=\{u=0,u_x=0\}$ which describes total
degeneracy of the $N\times m$ Jacobian matrix $u_x.$ Since ${\rm
codim}\ S= (N+1)m>m+1$, this singularity does not occur, for generic initial
conditions $u_0.$ Lyapunov-Schmidt reduction, based on rank $u_x\geq
1,$ then reduces the discussion of all singularities to the case $\dim
u=N-1=1$ and $\dim x=m-1$ which was already discussed above.

For example Figure \ref{Fig4.3}(a), now corresponding to $N=2,\;m=2,$
describes the coalescence and subsequent annihilation of a pair of
spiral tips. Note how the invariance of Brouwer degree forces the two
annihilating spiral tips to have opposite degree (\ref{4.1.4}). For the two
annihilating spirals this implies opposite rotation direction, in the
Winfree paradigm. See also Figure \ref{Fig4.6} for a numerical
simulation. It is noteworthy that the opposite phenomenon of
persistent spiral pair creation, Figure \ref{Fig4.3}(b), has not been observed,
to our knowledge, neither numerically nor experimentally.  This
unobserved effect cannot be excluded by comparison or genericity
principles.

\subsubsection{Collisions of scroll wave filaments}

Let us next apply the same reduction to the case $N=2,\;m=3$ of scroll
filament collision. By the above arguments, Lyapunov-Schmidt reduces
this case to $N=1,\;m=2.$ The annihilation for $t\nearrow t_0$
according to Figure \ref{Fig4.3}(a), now of a circular filament
$z(t,\cdot),$ has been observed. It corresponds to the collapse and
disappearance of a scroll ring. Note that the collapsing scroll
filament has to be non-twisted, due to Proposition \ref{p-4.1}.
Again, the antagonistic case of scroll ring creation analogous to
Figure \ref{Fig4.3}(b) has mysteriously escaped both numerical and
experimental documentation.

The only remaining singularity for $N=2,\;m=3$ alias $N=1,m=2$ is the
case of indefinite (reduced) Hessian as depicted in
Figure \ref{Fig4.4}. Note how the entire dynamic process of approach
(a), kiss (b), and reconnection (c) occurs in a single tangent
plane. Moreover, Figure \ref{Fig4.4} gives the definitive answer to the
opening question of this section: for generic initial conditions
$u_0\in X\hookrightarrow C^2,$ the topology changes of scroll wave
filaments must occur by crossover collisions following the kissing
scenario of Figure \ref{Fig4.2}(b).

We summarize these results as follows.

\begin{Corollary}\label{c-4.3}
\cite{fieman00} For generic initial conditions $u_0\in X\hookrightarrow
C^2,$ only the following three collision scenarios of scroll wave
filaments, as defined in (\ref{4.1.5}) above, are possible for $t>0$, $x\in\Omega$:
\begin{itemize}\item[(i)] annihilation of untwisted filaments,
according to Figure \ref{Fig4.3}(a);
\item[(ii)] creation of untwisted filaments, according to Figure
\ref{Fig4.3}(b); 
\item[(iii)] crossover collision of filaments, according to  Figures
\ref{Fig4.2}(b) and  \ref{Fig4.4}
\end{itemize}
\end{Corollary}

We emphasize that this result applies only in the interior of the
domain $\Omega$. We do not investigate, for example, the collision of
filament points in the boundary $\Omega.$

As another corollary, we note that 
%
\begin{eqnarray}\sum_j {\rm\, twist} (z_j) +2 \sum_{j<k}{\rm\, lk\;} (z_j,z_k) = const. \label{4.2.10}\end{eqnarray}
remains constant, for generic initial conditions $u_0\in X$,
independently of the time evolution of the closed filaments
$z_j=z_j(t,\cdot)$ and of possible changes in their number.
Mutual collision, or self-collision, of closed filaments may
change the total number of closed filaments but not the total
twist and linking according to (\ref{4.2.10}).
It is again assumed here --- similarly to degree theory --- that the
closed filaments $z_j\subset \Omega$ neither collide with the boundary
$\partial\Omega$ nor with filaments $z$ which extend to the
boundary. 

For a proof we consider the accompanying filaments
$\tilde{z}_j\subseteq u(t,\cdot)^{-1} (\delta,0)$ which define the
local phase along the scroll filament $z_j.$ Then
\begin{equation}\label{e-end1}
\displaystyle{\sum_j} {\rm\, twist} (z_j) + 2
\displaystyle{\sum_{j<k}}{\rm\, lk\;} (z_j,z_k) = 
\displaystyle{\sum_{j,k}} {\rm\, lk\;} (z_j,\tilde{z}_k),
\end{equation}
for small enough $\delta <0.$ None of the singularities (i)--(iii) of
Corollary \ref{c-4.3} changes the latter quantity.

If all filaments are closed, so that $u\neq 0$ on $\partial\Omega\cong
S^2,$ then we note in passing that
\begin{equation}\label{e-end2}
\sum_{j,k}{\rm lk\;} (z_j,\tilde{z}_k) = 0,
\end{equation}
by direct homotopy $u(t,\cdot)\rightarrow const. \neq 0.$

For an example, we recall Figure \ref{Fig4.1.2}. Another theoretical
example well worth contemplating is the clover-shaped trefoil knot as
shown in Figure \ref{Fig4.5}.  According to corollary (\ref{4.2.10})
the initial torus knot must possess zero twist. One way to see this,
directly, relies on the successive construction of {\em Seifert
surfaces} $S$: these are compact, orientable, embedded surfaces of minimal genus, with
the link, or knot, as boundary. We may interpret $S$ as
$u(t;\cdot)^{-1}([0,\infty)\times\{0\})$. We start with an embedded
disk spanning the circle, in the lower right of Figure
\ref{Fig4.5}. Crossover collisions change the topology of $S,$ but in
the present example $S$ will still qualify as a Seifert surface, both
before and after collision. In the resulting Seifert surface of the
original torus knot $z$, in the upper left of Figure \ref{Fig4.5}, our
interpretation produces $\tilde{z}=z(t;)^{-1}((\delta,0)).$ Direct
inspection then confirms zero twist of the torus knot, ${\rm twist}
(z)=\,{\rm lk\;}(z,\tilde{z})=0,$ as predicted by
(\ref{4.2.10}). Similarly (\ref{4.2.10}) shows that any single knot is
untwisted. Indeed any single knot can be decomposed into unlinked
planar circles, under a suitable sequence of crossover collisions. Any
twisted knot, in contrast, must be linked to another filament, again
by (\ref{4.2.10}). If two knots are mutually linked, but isolated
otherwise, then at least one of the knots must be twisted. These
remarks generalize our previous conclusions from proposition 6.1. For
a numerical example see Section \ref{s-4.3}.

%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,4.5)%
\centerline{\includegraphics[scale=0.8]{Fig/Fig29.eps}}
\end{picture}%
\caption{Schematic two-step disintegration of a torus knot filament via linked filaments.}
\label{Fig4.5}
\end{figure}


\subsection{Numerical simulations}
\label{s-4.3}
In the previous two sections we have presented mathematically rigorous
results concerning degrees, orientations, twists, linking, and
collisions of zeros of solutions $u=u(t,x)$ of reaction-diffusion
systems. The terminology of spiral tip, scroll waves, scroll wave
filaments, etc. was largely motivated by experimental observations and
numerical simulations.  We present some such simulations in the
present section. All results presented here are based on
\cite{fieman00}. The online version contains downloadable animations, for
the entertainment of our readers! As a model equation we consider the
FitzHugh-Nagumo system (\ref{4.1.1}), (\ref{4.1.2}). For parameters we
choose
%
\begin{eqnarray}\begin{array}{rcl} a &=& 0.8;\\
b &=& 0.01 + A\cos \omega t \quad {\rm with}\\
&& A = 0 {\rm\;or\;}0.01;\;\omega=3.21;\\
D &=& \left( \begin{array}{rc} 1 & 0\\ 0 & 0.5\end{array} \right);\\
\varepsilon &=& 0.02;\\
\Omega &=& [-15,15]^3;
\end{array} \label{4.3.1}\end{eqnarray}
unless stated otherwise. To define tip and filament positions, we use
the shifted reference values $u_1=u_2=0.5$ rather than $u=0.$ The
choices of $A$ correspond to autonomous versus slightly periodically
forced simulations.

For integration we use Barkley's code \cite{domaba97} with equidistant
$x$-discretizati\-on by $125^3$ points and a time step $\Delta
t=0.0172\cdots$. Filament positions are determined from the discrete
data by a simplex method in the spirit of \cite{allgeo90},
compatible with the above degree and orientation approach. To
visualize the local phase, as well as accompanying twist curves
$\tilde{z},$ we compute and plot a piecewise linear approximation to
the surface in $\Omega$ given by the preimage under $u(t,\cdot)$ of
the band
%
\begin{eqnarray}u_2 = 0.5 \leq u_1 \leq 0.5+\delta. \label{4.2.12}\end{eqnarray}
Following the Winfree paradigm, we call this surface with boundaries
$z$ and $\tilde{z}$ the {\em isochrone band.}

The choice of initial conditions which produce filament collisions can
be a delicate issue.  For planar simulations, $m=\dim x=2,$ it is
relatively easy to prescribe ``vector fields'' $u_0=u_0(x)$ with zeros
of prescribed degree. The reaction-diffusion system then quickly
produces spirals with tips, near these zeros, and with rotation
directions according to local Brouwer degree. Collisions can then be
produced, with a little bit of forcing. For three-dimensional
simulations, \cite{henwin91} has described a method which can produce linked
circular filaments and torus knots. This method is based on algebraic
topology: the initial filament configurations in $x\in\Omega$ are
realized by zeros $z$ of appropriate complex polynomials
$u(x)=p(z)\in\mathc\cong\mathr^2,$ in two complex variables
$z=(z_1,z_2).$ By inverse stereographic projection $\iota$, we have
$x\in \Omega\subset \mathr^3 \stackrel{\iota}{\hookrightarrow}
S^3\subset \mathc^2$ corresponding to the argument $z$ of $p(z)$.

We now describe the individual runs. The planar simulation of Figure
\ref{Fig4.6} shows color coded solution profiles $u_1(t,\cdot)$
corresponding to two meandering spirals which approach each other and
annihilate. The domain $\Omega=[-15,15]^2$ was discretized by $125^2$
points.  Slightly differently from Section~\ref{s:meander}, the
meandering motion is produced by an external time-periodic, spatially
homogeneous forcing $A=0.01.$ The frequency $\omega=3.21$ and the
phase were tuned to near drift resonance, so that the two spirals are
moving towards each other, initially. Rather than mutual repulsion,
annihilation occurs. Note the meandering, near drifting behavior of
the spiral tips $u_1=u_2=0.5,$ traced in white.

%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\begin{picture}(13.5,6.)%
\centerline{\includegraphics[scale=0.35]{Fig/30.eps}}
\end{picture}%
\caption{\cite[Fig. 7]{fieman00} Meandering interaction and collision of a  pair of planar spiral
waves.} \label{Fig4.6}
\end{figure}


%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\centerline{\includegraphics[scale=0.3]{Fig/31a.eps}}
\centerline{\includegraphics[scale=0.3]{Fig/31b.eps}}

\centerline{\includegraphics[scale=0.3]{Fig/31c.eps}}
\caption{\cite[Fig. 8]{fieman00} Scroll ring annihilation of an untwisted filament. Surfaces $u_2=0\leq u_1$
shown.} \label{Fig4.7}
\end{figure}


%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\centerline{\includegraphics[scale=0.25]{Fig/32a.eps}}
\centerline{\includegraphics[scale=0.25]{Fig/32b.eps}}

\centerline{\includegraphics[scale=0.25]{Fig/32c.eps}}
\centerline{\includegraphics[scale=0.25]{Fig/32d.eps}}
\caption{\cite[Fig. 9]{fieman00} Crossover collision of scroll waves including
accompanying isochrone bands $u_2=0\leq u_1\leq \delta.$}
\label{Fig4.8}
\end{figure}

%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\centerline{\includegraphics[scale=0.25]{Fig/33a.eps}}
\centerline{\includegraphics[scale=0.25]{Fig/33b.eps}}

\centerline{\includegraphics[scale=0.25]{Fig/33c.eps}}
\caption{\cite[Fig. 11]{fieman00} Crossover collision of two linked, twisted, circular
filaments.} 
\label{Fig4.9}
\end{figure}

%
\begin{figure}%[t]
\setlength{\unitlength}{1cm}%
\centerline{\includegraphics[scale=0.2]{Fig/34a.eps}}%
\centerline{\includegraphics[scale=0.2]{Fig/34b.eps}}

\centerline{\includegraphics[scale=0.2]{Fig/34c.eps}}
\centerline{\includegraphics[scale=0.3]{Fig/34d.eps}}
\caption{\cite[Fig. 13]{fieman00} Decomposition of the trefoil knot
into two linked, twisted circular filaments.} \label{Fig4.10}
\end{figure}


The annihilation of a three-dimensional untwisted scroll ring in
Figure \ref{Fig4.7} is an autonomous simulation, $A=0.$ Note the analogy
to the planar simulation of spiral pair annihilation, by viewing a
planar vertical slice through the toroidal scroll ring. The
transparent full isochrone surfaces in $\Omega=[-15,15]^3$ given by
$u_2=0.5\leq u_1$ and shown in this example would obscure the view on
the scroll filaments $u_2=0.5=u_1$ in more complicated
simulations. Therefore they are replaced by the truncated isochrone
bands $u_2=0.5\leq u_1 \leq 0.5+\delta,$ as was explained above.

Crossover collision is documented in Figure \ref{Fig4.8}. For finer
resolution we choose a smaller domain $\Omega=[-10,10]^3,$ again with
$125^3$ discretization points. The appropriately adjusted time step is
$\Delta t= 0.00765.$ We choose two straight line horizontal filaments,
orthogonal to each other and at a vertical distance. By a spatially
homogeneous, time periodic forcing $A=0.01,$ as before, we can make
the top filament drift downwards and the bottom filament drift
upwards. Because the break through scenario of Figure \ref{Fig4.2}(a)
is forbidden, generically, the filaments have to first locate a
tangent plane for kissing.  Indeed, the crossover collision then
occurs between $t=35$ and $t=40.$

Based on the polynomial $z(z)=z_1^2-z^2_2,$ an unlinking collision of
two linked, twisted, circular scroll filaments is shown in Figure
\ref{Fig4.9}. The example is autonomous, $A=0,$ with data as in
(\ref{4.3.1}). Again a common tangent plane is located. Kissing and
reconnection according to the abstract scenario of Figures \ref{Fig4.1.2}, 
\ref{Fig4.2}(b), and \ref{Fig4.4} occurs between $t=4.8$ and $t=4.9.$

Our final simulation, the decomposition of a trefoil knot into two
linked, twisted, circular scroll filaments, runs on a spatially
coarser $125^3$ grid on $\Omega=[-25,25]^3.$ The construction of the
initial condition is based on the polynomial $p(z)=z^2_1-z^3_2.$ The
autonomous run, $A=0,$ follows the theoretical scenario of Figures
\ref{Fig4.2}(b) and \ref{Fig4.5}, fully compliant with our abstract theory.

Our theoretical results apply to any fixed value $u=c$ and the
associated level curves of the spatial profiles. It is the prominence
of the particular values associated to spiral tips, however, becoming
visible only through experimental work and numerical simulation along
the Winfree paradigm, which has motivated our mathematical analysis.

\begin{thebibliography}{20}
\bibitem[AbRo67]{abrrob67}
R.~Abraham and J.~Robbin.
 {\em Transversal Mappings and Flows}.
 Benjamin Inc., Amsterdam, 1967.
\bibitem[Al79]{alikik79}
N.~Alikakos.
 An application of the invariance principle to reaction diffusion
  equations.
 {\em J. Diff. Eqns.} {\bf 33} (1979), 201--225.
\bibitem[Al\&al89]{albafu89}
N.~Alikakos, P.W. Bates, and G.~Fusco.
 Slow motion for the {C}ahn-{H}illiard equation in one space
  dimension.
{\em  Preprint} (1989).
\bibitem[AlGe90]{allgeo90}
E.L. Allgower and K.~Georg.
 {\em Numerical Continuation Methods. An Introduction}.
 Springer-Verlag, Berlin, 1990.
\bibitem[An86]{angene86b}
S.~Angenent.
 The {M}orse-{S}male property for a semi-linear parabolic equation.
 {\em J.~Diff.~Eqns.} {\bf 62} (1986), 427--442.
\bibitem[An88]{angene88}
S.~Angenent.
 The zero set of a solution of a parabolic equation.
 {\em Crelle J.~reine~angew.~Math.}, {\bf 390} (1988), 79--96.
\bibitem[An90]{angene90}
S.~Angenent.
 Parabolic equations for curves on surfaces. {I:} curves with
  $p$-integrable curvature.
 {\em Ann. Math.} {\bf 132} (1990), 451--483.
\bibitem[An91]{angene91}
S.~Angenent.
 Parabolic equations for curves on surfaces. {II}: Intersections,
  blow-up and generalized solutions.
 {\em Ann. Math.}, {\bf 133} (1991), 171--215.
\bibitem[An93]{angene93}
S.~Angenent.
 A variational interpretation of {M}elnikov's function and
  exponentially small separatrix splitting.
 {\em Lond. Math. Soc. Lect. Note Ser.}, {\bf 192} (1993), 5--35.
\bibitem[AnFi88]{angfie88}
S.~Angenent and B.~Fiedler.
 The dynamics of rotating waves in scalar reaction diffusion
  equations.
 {\em Trans.~Amer.~Math.~Soc.}, {\bf 307} (1988), 545--568.
\bibitem[An\&al87]{anmape87}
S.~Angenent, J.~Mallet-Paret, and L.A. Peletier.
 Stable transition layers in a semilinear boundary value problem.
 {\em J.~Diff.~Eqns.} {\bf 67} (1987), 212--242.
\bibitem[An91]{anosov91}
D.V. Anosov.
{\em Dynamical Systems with Hyperbolic Behaviour}.
Enc. Math. Sc. {\bf 66}, Dynamical Systems {IX}. Springer-Verlag, New
York, 1991.
\bibitem[AnAr88]{anoarn88} D. V. Anosov and V. I. Arnol'd. {\em
Ordinary differential equations and smooth dynamical systems.}
Enc. Math. Sc. {\bf 1}, Dynamical Systems {I}. 
Springer-Verlag, Berlin, 1988. 
\bibitem[Ar\&al92]{Kramer}
I.~S.~Aranson, L.~Aranson, L.~Kramer, and A.~Weber.
Stability limits of spirals and travelling waves in nonequilibrium media.
{\it Phys. Rev.~A} {\bf 46} (1992), 2992--2995.
\bibitem[Ar\&al94]{AKW}
I.S. Aranson, L. Kramer, and A. Weber.
 Core instability and spatiotemporal intermittency of spiral waves in
 oscillatory media.
{\em Phys. Rev. Lett.} {\bf 72}, 2316 (1994).
\bibitem[Ar92]{arnold92}
V.I. Arnol'd.
 {\em Theory of Bifurcations and Catastrophes}.
 Enc. Math. Sc. {\bf 5}, Dynamical Systems {V}. Springer-Verlag,
  Berlin, 1992.
\bibitem[Ar93]{arnold93}
V.I. Arnol'd.
 {\em Singularity Theory I}.
 Enc. Math. Sc. {\bf 6}, Dynamical Systems {VI}. Springer-Verlag, New
  York, 1993.
\bibitem[Ar94a]{arnold94}
V.I. Arnol'd.
 {\em Bifurcation Theory and Catastrophe Theory}.
 Enc. Math. Sc. {\bf 5}, Dynamical Systems {V}. Springer-Verlag, New York,
  1994.
\bibitem[Ar94b]{arnold94b} V.I. Arnol'd.  
{\em Singularity
theory II, Applications}.  Enc. Math. Sc. {\bf 8}, Dynamical Systems
{VIII}. Springer-Verlag, New York, 1993. 
\bibitem[Ar\&al85]{arguva85}
V.I. Arnol'd, S.M. Gusejn-Zade, and A.N. Varchenko.
 {\em Singularities of Differentiable Maps. Volume I: The
  Classification of Critical points, Caustics and Wave Fronts}.
 Birkh{\"a}user, Boston, 1985.
\bibitem[Ar\&al88]{arnoetal88} V.I. Arnol'd, V.V. Kozlov, and
A.I. Neishtadt.  {\em Mathematical Aspects of Classical and Celestial
Mechanics}.  Enc. Math. Sc. {\bf 3}, Dynamical Systems
{III}. Springer-Verlag, New York, 1988.
\bibitem[ArNo90]{arnnov90}
V.I. Arnol'd and S.P. Novikov.
 {\em Symplectic Geometry and its Applications}.
 Enc. Math. Sc. {\bf 4}, Dynamical Systems {IV}. Springer-Verlag, New
  York, 1990.
\bibitem[ArNo94]{arnnov94}
V.I. Arnol'd and S.P. Novikov.
{\em Integrable Systems. Nonholonomic Dynamical Systems}.
Enc. Math. Sc. {\bf 16}, Dynamical Systems {VII}. Springer-Verlag, New
York, 1994.
\bibitem[ArVi89]{arnvis89}
V.I. Arnol'd and M.I.~Vishik et~al.
Some solved and unsolved problems in the theory of differential
equations and mathematical physics.
{\em Russian Math.~Surveys}, {\bf 44} (1989), 157--171.
\bibitem[AM97]{ashmel97}
P.~Ashwin and I.~Melbourne. 
Noncompact drift for relative equilibria and relative periodic orbits.
{\em Nonlinearity}, {\bf 10} (1997), 595--616.
\bibitem[As\&al99]{AshMel} P. Ashwin, I. Melbourne, and M. Nicol.
Drift bifurcations of relative equilibria and transitions of
spiral waves.  
{\em Nonlinearity} {\bf 12} (1999),  741--755.
\bibitem[As\&al01]{asmeni01} P. Ashwin, I. Melbourne, and M. Nicol. 
Hypermeander of spirals: local bifurcations and statistical
properties.  {\em Phys. D} {\bf 156} (2001), 364--382.
\bibitem[BaVi92]{babvis92}
A.V. Babin and M.I. Vishik.
 {\em Attractors of Evolution Equations}.
 North Holland, Amsterdam, 1992.
\bibitem[B{\"a}Ei93]{BE}
M. B\"ar and M. Eiswirth.
Turbulence due to spiral breakup in a continuous excitable medium.
{\em Phys. Rev. E} {\bf 48} (1993), 1635--1637.
\bibitem[B{\"a}Or99]{BO}
M. B\"ar and M. Or-Guil.
Alternative scenarios of spiral breakup in a reaction-diffusion model
with excitable and oscillatory dynamics.
{\em Phys. Rev. Lett.} {\bf 82} (1999), 1160--1163.
\bibitem[Ba92]{Barkley92}
D.~Barkley. Linear stability analysis of rotating spiral waves in
excitable media.
{\it Phys. Rev. Lett.} {\bf 68} (1992), 2090--2093.
\bibitem[Ba93]{Barkley93}
D.~Barkley.
Euclidean symmetry and the dynamics of rotating spiral waves.
{\it Phys. Rev. Lett.} {\bf 72} (1994), 164--167.
\bibitem[Bar95]{barkle95}
D.~Barkley.
Spiral meandering.
 {I}n R. Kapral and K. Showalter (eds.), {\em {C}hemical {W}aves and
  {P}atterns, p.163--190}, Kluwer, 1995.
\bibitem[Ba\&al93]{barlesetal93}
G.~Barles, H.M. Soner, and P.E. Souganidis.
 Front propagation and phase field theory.
 {\em SIAM J. Contr. Optim.} {\bf 31} (1993), 439--469.
\bibitem[Be\&al97]{OF} A. Belmonte, J.-M. Flesselles, and
Q. Ouyang. Experimental Survey of Spiral Dynamics in the
Belousov-Zhabotinsky Reaction.  {\em J. Physique II} {\bf 7} (1997),
1425--1468.
\bibitem[BeNi90]{BeNi} H.\,Berestycki and L.\,Nirenberg. Some qualitative
properties of solutions of semilinear elliptic equations in
cylindrical domains. {\it  Coll.\,Analysis, et cetera}, 115--164, Academic
Press Boston, 1990.
\bibitem[BePh96]{BePh96} I.~Berkes and W.~Philipp.
Trigonometric series and uniform distribution $\mbox{mod }1$.
Stud. Sci. Math. Hung. {\bf 31} (1996), 15--25.
\bibitem[Be87]{beyn87}
W.J. Beyn.
 The effect of discretization on homoclinic orbits.
 In {\em Bifurcation: Analysis, Algorithms, Applications 1--8, {\em T.
  K{\"u}pper et al., (eds.)}}. Birkh{\"a}user Verlag, Basel, 1987.
\bibitem[Be90]{beyn90}
W.-J. Beyn.
 The numerical computation of connecting orbits in dynamical systems.
 {\em IMA Z. Numer. Anal}, {\bf 9} (1990), 379--405.
\bibitem[Bi\&al96]{hoetal96}
V.~A. Biktashev, A.~V. Holden, and E.~V. Nikolaev.
 Spiral wave meander and symmetry of the plane.
 Preprint, University of Leeds, 1996.
\bibitem[BiRo62]{birrot62}
G.~Birkhoff and G.-C. Rota.
 {\em Ordinary differential equations}.
 Ginn and Company, Boston, 1962.
\bibitem[Bl\&al00]{deKepper} P. Blancheau, J. Boissonade, and P. De Kepper.
Theoretical and experimental studies of bistability in the
chloride-dioxide-iodide reaction. {\em Physica D} {\bf 147} (2000), 283--299.
\bibitem[Bo81a]{bogdan81a}
R.~Bogdanov.
 Bifurcation of the limit cycle of a family of plane vector fields.
 {\em Sel.~Mat.~Sov.} {\bf 1} (1981), 373--387.
\bibitem[Bo81b]{bogdan81b}
R.~Bogdanov.
 Versal deformations of a singularity of a vector field on the plane
  in the case of zero eigenvalues.
 {\em Sel. Mat. Sov.}, {\bf 1} (1981), 389--421.
\bibitem[BrEn93]{braeng93}
M.~Braune and H.~Engel.
Compound rotation of spiral waves in a light-sensitive
Belousov-Zhabotinsky medium.
{\em Chem. Phys. Lett.} {\bf 204} (1993), 257--264.
\bibitem[Br64]{Briggs}
R. J. Briggs.
{\em Electron-Steam Interaction With Plasmas}.
MIT press, Cambridge, 1964.
\bibitem[BrtD85]{brotdi85}
T.~Br{\"o}cker and T.~tom Dieck.
 {\em Representations of Compact Lie Groups}.
 Springer-Verlag, Berlin, 1985.
\bibitem[Br\&al01]{brkrve01}
H.W. Broer, B.~Krauskopf, and G.~Vegter.
 {\em Global Analysis of Dynamical Systems}.
 IOP Publishing, Bristol, 2001.
\bibitem[BrTa02]{broer} H.~Broer and T.~Takens (eds.). {\em {H}andbook
of {D}ynamical {S}ystems} {\bf 3}.  {E}lsevier, {A}msterdam, in preparation 2002.
\bibitem[Br90]{brunov90}
P.~Brunovsk\'y.
The attracor of the scalar reaction diffusion equation is a smooth
graph.
{\em J.~Dynamics and Differential Equations}, {\bf 2}
(1990), 293--323.
\bibitem[BrCh84]{brucho84}
P.~Brunovsk\'y and S-N Chow.
Generic properties of stationary state solutions of
reaction-diffusion equations.
{\em J. Diff. Eqns.} {\bf 53} (1984), 1--23.
\bibitem[BrFi86]{brufie86a}
P.~Brunovsk\'y and B.~Fiedler.
 Numbers of zeros on invariant manifolds in reaction-diffusion
  equations.
 {\em Nonlin.~Analysis, TMA}, {\bf 10} (1986), 179--194.
\bibitem[BrFi88]{brufie88}
P.~Brunovsk\'y and B.~Fiedler.
 Connecting orbits in scalar reaction diffusion equations.
 {\em Dynamics Reported} {\bf 1} (1988), 57--89.
\bibitem[BrFi89]{brufie89}
P.~Brunovsk\'y and B.~Fiedler.
 Connecting orbits in scalar reaction diffusion equations {II}: The
  complete solution.
 {\em J.~Diff.~Eqns.} {\bf 81} (1989), 106--135.
\bibitem[Br\&al92]{brposa92}
P.~Brunovsk\'y, P.~Pol\'a\v{c}ik, and B.~Sandstede.
 Convergence in general parabolic equations in one space dimension.
 {\em Nonl.~Analysis~TMA} {\bf 18} (1992), 209--215.
\bibitem[Ca\&al93]{CaMoSo} A. Calsina, X. Mora and J. Sol\`{a}-Morales.
The dynamical approach to elliptic problems in cylindrical domains 
and a study of their parabolic singular limit. {\it  J. Diff. Eqns.} {\bf
102} (1993), 244--304.
\bibitem[CaPe90]{Car} J. Carr and R. Pego.Invariant manifolds for metastable
patterns in $u\sb t=\epsilon\sp 2u\sb {xx}-f(u)$.
 {\em Proc. Roy. Soc. Edinburgh A} {\bf 116} (1990), 133--160.
\bibitem[Ca\&al90]{firstturing} 
V. Castets, E. Dulos, J. Boissonade, and P. De Kepper.
 Experimental evidence of a sustained standing Turing-type
nonequilibrium chemical pattern.
{\em Phys. Rev. Lett.} {\bf 64} (1990), 2953--2956.
\bibitem[ChIn74]{chainf74}
N.~Chafee and E.~Infante.
 A bifurcation problem for a nonlinear parabolic equation.
 {\em J.~Applicable Analysis} {\bf 4} (1974). 17--37.
\bibitem[Ch98]{Chen}
X.-Y. Chen.
 A strong unique continuation theorem for parabolic equations.
{\it Math. Ann.} {\bf 311} (1998), 603--630.
\bibitem[ChVi02]{chevis02}
V.V. Chepyzhov and M.I. Vishik.
 {\em Attractors for {E}quations of {M}athematical {P}hysics}.
 Colloq. AMS, Providence, 2002.
\bibitem[ChLa00]{cholau00}
P.~Chossat and R.~Lauterbach.
 {\em Methods in Equivariant Bifurcations and Dynamical Systems}.
 World Scientific, Singapore, 2000.
\bibitem[ChHa82]{chowhal82}
S.-N. Chow and J.~K. Hale.
 {\em Methods of Bifurcation Theory}.
 Springer-Verlag, New York, 1982.
\bibitem[CoEc00]{CoEc} P. Collet and J.-P. Eckmann. Proof of the marginal
stability bound for the Swift-Hohenberg equation and related
equations. {\em  Preprint}, 2000.
\bibitem[Co68]{coppel68}
W.A. Coppel.
 Dichotomies and reducibility {II}.
 {\em J.~Diff. Eqns.} {\bf 4} (1968), 386--398.
\bibitem[Co78]{Coppel}
W.A. Coppel.
{\em Dichotomies in Stability Theory}.
Lect. Notes Math. {\bf 629}, Springer, Berlin, 1978.
\bibitem[CrHo93]{CH} M.C. Cross and P.C. Hohenberg. 
Pattern formation outside equilibrium.
{\em Rev. Modern Phys}. {\bf 65} (1993), 851--1112.
\bibitem[Dam97]{damon97}
J.~Damon.
 Generic properties of solutions to partial differential equations.
 {\em Arch. Rat. Mech. Analysis}, {\bf 140} (1997), 353--403.
\bibitem[DaPo02]{dapo02} E.N. Dancer and P. Pol\'a\v{c}ik. 
{\em Realization of vector fields and dynamics of spatially
homogeneous parabolic equations.}  Mem. AMS, Providence, 2002, to appear.
\bibitem[Da\&al96]{dafiki96}
G.~Dangelmayr, B.~Fiedler, K.~Kirchg{\"a}ssner, and A.~Mielke.
 {\em Dynamics of Nonlinear Waves in Dissipative Systems: Reduction,
  Bifurcation and Stability}.
 Pitman {\bf 352}, Boston, 1996.
\bibitem[De\&al95]{DGHS} M. Dellnitz, M. Golubitsky, A. Hohmann,and
I. Stewart. Spirals in scalar reaction-diffusion equations. {\em
Internat. J. Bifur. Chaos Appl. Sci.  Engrg.} {\bf 5} (1995),
1487--1501.
\bibitem[De85]{deimli85}
K.~Deimling.
 {\em Nonlinear Functional Analysis}.
 Springer-Verlag, Berlin, 1985.
\bibitem[dK\&al94]{dekepper} P. de Kepper, J.-J. Perraud,
B. Rudovics,and E. Dulos.  Experimental study of stationary Turing
patterns and their interaction with traveling waves in a chemical
system.  {\em Int. J. Bifurcation Chaos Appl. Sci. Eng.} {\bf 4}
(1994), 1215-1231.
\bibitem[Di\&al95]{dieketal95}
O.~Diekmann, S.A. v.~Gils, S.M.~Verduyn Lunel, and H.-O. Walther.
 {\em Delay Equations. Functional-, Complex-, and Nonlinear Analysis}.
 Springer-Verlag, New York, 1995.
\bibitem[DoFr89]{doefri89}
E.~J. Doedel and M.~J. Friedman.
 Numerical computation of heteroclinic orbits.
 {\em J.~Comp. Appl.~Math.}{\bf 26} (1989), 155--170.
\bibitem[DoFr91]{doefri91}
E.~J. Doedel and M.~J. Friedman.
 Numerical computation and continuation of invariant manifolds
  connecting fixed points.
 {\em SIAM J. Numer. Anal.} {\bf 28} (1991), 789--808.
\bibitem[Do\&al97]{domaba97}
M.~Dowle, M.~Mantel, and D.~Barkley.
 Fast simulations of waves in three-dimensional excitable media.
 {\em Int. J. Bifur. Chaos}, {\bf 7} (1997), 2529--2546.
\bibitem[Du\&al85]{dufono85}
B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov.
 {\em Modern Geometry - Methods and Applications. Part 2: The Geometry
  and Topology of Manifolds}.
 Springer-Verlag, New York, 1985.
\bibitem[EiYa98]{eiyan98}
S.-I. Ei and E.~Yanagida.
 Slow dynamics of interfaces in the allen-cahn equation on a
  strip-like domain.
 {\em SIAM J. Math. Anal.}, {\bf 29} (1998), 555--595.
\bibitem[El\&al87]{elphetal87}
C. Elphick, E. Tirapegui, M.E. Brachet,
P. Coullet, and G. Iooss.
 A simple global characerization for normal forms of singular vector
  fields.
 {\em Physica} {\bf 29D} (1987), 95--127.
\bibitem[Fe71]{feniche71}
N.~Fenichel.
 Persistence and smoothness of invariant manifolds for flows.
 {\em Indiana Univ.~Math.~J.} {\bf 21} (1971), 193--226.
\bibitem[Fe74]{feniche74}
N.~Fenichel.
 Asymptotic stability with rate conditions.
 {\em Indiana Univ.~Math.~J.} {\bf 23} (1974), 1109--1137.
\bibitem[Fe77]{feniche77}
N.~Fenichel.
 Asymptotic stability with rate conditions, {II}.
 {\em Indiana Univ.~Math.~J.} {\bf 26} (1977), 81--93.
\bibitem[Fe79]{feniche79}
N.~Fenichel.
 Geometric singular perturbation theory for ordinary differential
  equations.
 {\em J. Diff.~Eqns.}, {\bf 31} (1979), 53--98.
\bibitem[Fi88]{fiedle88}
B.~Fiedler.
 {\em Global Bifurcation of Periodic Solutions with Symmetry}.
 Springer-Verlag, Berlin, 1988.
\bibitem[Fi89]{fiedle89}
B.~Fiedler.
 Discrete {L}japunov functionals and $\omega$-limit sets.
 {\em Math. Mod. Num. Analysis}, {\bf 23} (1989), 415--431.
\bibitem[Fi94]{fiedle94}
B.~Fiedler.
 Global attractors of one-dimensional parabolic equations: sixteen
  examples.
 {\em Tatra Mountains Math. Publ.}, {\bf 4} (1994), 67--92.
\bibitem[Fi96]{Fie7} 
B. Fiedler.
Do global attractors depend on boundary conditions? 
{\em Doc. Math.} {\bf 1} (1996), 215--228.
\bibitem[Fi02]{fiedlerhand} B. Fiedler (ed.)  {\em {H}andbook of {D}ynamical
{S}ystems} {\bf 2}, 
{E}lsevier, {A}msterdam. In press.
\bibitem[FiGe98]{fieged98}
B.~Fiedler and T.~Gedeon.
 A class of convergent neural network dynamics.
 {\em Physica D}, {\bf 111} (1998), 288--294,.
\bibitem[FiGe99]{fieged99} B.~Fiedler and T.~Gedeon.  A {L}yapunov
function for tridiagonal competitive-cooperative systems.  {\em SIAM
J. Math Analysis} {\bf 30} (1999), 469--478.
\bibitem[Fi\&al00]{figrsp98} B.~Fiedler, K. Gr\"oger, and J. Sprekels
(eds.).  {\em Equadiff 99. International Conference on Differential
Equations, Berlin 1999}. Vol.1,2. World Scientific, Singapore, 2000.
\bibitem[FiMP89a]{fiemal89a}
B.~Fiedler and J.~Mallet-Paret.
 Connections between {M}orse sets for delay-differential equations.
 {\em J. reine angew. Math.}, {\bf 397}:23--41, (1989).
\bibitem[FiMP89b]{fiemal89b}
B.~Fiedler and J.~Mallet-Paret.
 A {P}oincar\'e-{B}endixson theorem for scalar reaction diffusion
  equations.
 {\em Arch. Rat. Mech. Analysis} {\bf 107} (1989), 325--345.
\bibitem[FiMa00]{fieman00}
B.~Fiedler and R.-M. Mantel.
 Crossover collision of core filaments in three-dimensional scroll
  wave patterns.
 {\em Doc. Math.} {\bf 5} (2000), 695--731.
\bibitem[FiPo90]{fiepol90}
B.~Fiedler and P.~Pol\'{a}\v{c}ik.
 Complicated dynamics of scalar reaction diffusion equations with a
  nonlocal term.
 {\em Proc. Royal Soc. Edinburgh} {\bf 115A} (1990), 167--192.
\bibitem[FiRo96]{fieconstr} B. Fiedler and C. Rocha.  Heteroclinic orbits of
semilinear parabolic equations. {\em  J. Diff. Eq.} {\bf 125} (1996), 
239--281. 
\bibitem[FiRo99]{fieroc99} B.~Fiedler and C.~Rocha.
Realization of meander permutations by boundary value problems.  {\em
J. Diff. Eqns.} {\bf 156} (1999), 282--308.
\bibitem[FiRo00]{fieroc00}
B.~Fiedler and C.~Rocha.
 Orbit equivalence of global attractors of semilinear parabolic
  differential equations.
 {\em Trans. Amer. Math. Soc.}, {\bf 352} (2000), 257--284.
\bibitem[Fi\&al02a]{fienochmal} B. Fiedler, C. Rocha, D. Salazar, and
J. Solà-Morales.  
A note on the dynamics of piecewise-autonomous bistable parabolic
equations.   Comm. Fields Inst. (2002), in press.
\bibitem[Fi\&al02b]{firowo} B. Fiedler, C. Rocha, and M. Wolfrum. Heteroclinic connections of $S^1$-equivariant parabolic equations on the circle. In preparation, 2002. 
\bibitem[Fi\&al96]{FSSW} B. Fiedler, B. Sandstede, A. Scheel, and C.
Wulff. Bifurcation from relative equilibria of noncompact group
actions: skew products, meanders and drifts. {\it  Doc. Math. J. DMV} {\bf
1} (1996), 479--505. See also {\em
http://www.mathematik.uni-bielefeld.de/documenta/vol-01/20.ps.gz}
\bibitem[Fi\&al98]{fiscvi98}
B.~Fiedler, A.~Scheel, and M.~Vishik.
 Large patterns of elliptic systems in infinite cylinders.
 {\em J. Math. Pures Appl.} {\bf 77} (1998), 879--907.
\bibitem[FiTu98]{fietur98}
B.~Fiedler and D.~Turaev.
 Normal forms, resonances, and meandering tip motions near relative
  equilibria of {E}uclidean group actions.
 {\em Arch. Rat. Mech. Anal.} {\bf 145} (1998), 129--159.
\bibitem[FiVi01a]{fievis01b}
B.~Fiedler and M.~Vishik.
 Quantitative homogenization of analytic semigroups and reaction
  diffusion equations with diophantine spatial frequencies.
 {\em Adv. in Diff. Eqns.} {\bf 6} (2001), 1377--1408.
\bibitem[FiVi01b]{fievis01a}
B.~Fiedler and M.~Vishik.
 Quantitative homogenization of global attractors for
  reaction-diffusion systems with rapidly oscillating terms.
 {\em Preprint}, 2001.
\bibitem[Fi88]{fife88}
P.C. Fife.
 {\em Dynamics of internal layers and diffusive interfaces},
  {\em CBMS-NSF Reg. Conf. Ser. Appl. Math.} {\bf 53},
 1988.
\bibitem[Fi84]{fischer84}
G.~Fischer.
 Zentrumsmannigfaltigkeiten bei elliptischen {D}ifferentialgleichungen.
 {\em Math. Nachr.} {\bf 115} (1984), 137--157.
\bibitem[Fr64]{friedm64}
A.~Friedman.
 {\em Partial Differential Equations of Parabolic Type}.
 Prentice Hall, Englewood Cliffs, New Jersey, 1964.
\bibitem[FuHa89]{Fus} G. Fusco and J.K. Hale. Slow-motion manifolds,
dormant instability, and singular perturbations.
{\em J. Dyn. Diff. Eqns.} {\bf 1} (1989), 75--94.
\bibitem[FuOl88]{fusoli88}
G.~Fusco and W.M. Oliva.
 Jacobi matrices and transversality.
 {\em Proc. Royal Soc. Edinburgh {\bf A}} {\bf 109} (1988), 231--243.
\bibitem[FuRo91]{fusroc91}
G.~Fusco and C.~Rocha.
 A permutation related to the dynamics of a scalar parabolic {PDE}.
 {\em J. Diff. Eqns.} {\bf 91} (1991), 75--94.
\bibitem[GaHa86]{gagham86}
M.~Gage and R.S. Hamilton.
 The heat equation shrinking convex plane curves.
 {\em J. Diff. Geom.} {\bf 23} (1986), 69--96.
\bibitem[GaSl02]{gasl02}
T.~Gallay and S.~Slijep\v{c}evi\v{c}.
Personal communication, (2002).
\bibitem[GiHi96a]{giahil96a}
M.~Giaquinta and S.~Hildebrandt.
 {\em Calculus of Variations 1. The Lagrangian Formalism.}
 Springer-Verlag, Berlin, 1996.
\bibitem[GiHi96b]{giahil96b}
M.~Giaquinta and S.~Hildebrandt.
 {\em Calculus of Variations 2. The Hamiltonian Formalism}.
 Springer-Verlag, Berlin, 1996.
\bibitem[Go\&al00]{GKS} M. Golubitsky, E. Knobloch, and I. Stewart.  Target
patterns and spirals in planar reaction-diffusion
systems. {\em J. Nonlinear Sci.} {\bf 10} (2000), 333--354.
\bibitem[Go\&al97]{Gol} M.~Golubitsky, V.~LeBlanc, and I.~Melbourne. Meandering of the spiral tip: an alternative approach.  
J. Nonl. Sci. {\bf 7}
(1997), 557--586.
\bibitem[GoSc85]{golsch85}
M.~Golubitsky and D.G. Schaeffer.
 {\em {Singularities and Groups in Bifurcation Theory {I} }}.
 Springer-Verlag, 1985.
\bibitem[Go\&al88]{gostsc88}
M.~Golubitsky, I.~Stewart, and D.G. Schaeffer.
 {\em Singularities and Groups in Bifurcation Theory {II}}.
 Springer-Verlag, New York, 1988.
\bibitem[Go\&al98]{perioddoubling} 
A. Goryachev, H. Chat\'{e}, and R. Kapral.
Synchronization defects and broken symmetry in spiral waves.
{\em Phys. Rev. Lett.} \textbf{80} (1998), 873--876.
\bibitem[Gr89]{grayso89}
M.~A. Grayson.
 Shortening embedded curves.
 {\em Ann. Math.} {\bf 129} (1989), 71--111.
\bibitem[GuHo83]{guchol83}
J.~Guckenheimer and P.~Holmes.
 {\em Nonlinear Oscillations, Dynamical Systems, and Bifurcations of
  Vector Fields}.
 Springer-Verlag, New York, 1983.
\bibitem[Ha\&al]{haasetal95}
G.~Haas, M.~B{\"a}r, and I.G.~Kevrekidis et~al.
 Observation of front bifurcations in controlled geometries: From one
  to two dimensions.
 {\em Phys. Rev. Lett.} {\bf 75} (1995), 3560--3563.
\bibitem[Ha82]{Hagan}
P.S. Hagan.
Spiral waves in reaction-diffusion equations.
{\it SIAM J. Appl. Math.} {\bf 42} (1982), 762--786.
\bibitem[Ha69]{hale69}
J.K. Hale.
 {\em Ordinary Differential Equations}.
 John Wiley \& Sons, New York, 1969.
\bibitem[Ha85]{ha85}
J.K. Hale.
Flows on centre manifolds for scalar functional differential equations.
{\em Proc. R. Soc. Edinb., Sect. A} {\bf 101} (1985), 193--201.
 \bibitem[Ha88]{hale88}
J.K. Hale.
 {\em Asymptotic Behavior of Dissipative Systems}.
 Math. Surv. {\bf 25}. AMS Publications, Providence, 1988.
\bibitem[HaRa92]{hara92} 
J.K. Hale and G. Raugel.
Reaction-diffusion equation on thin domains. 
J. Math. Pures Appl. {\bf 71} (1992), 33--95.
\bibitem[H{\"a}97]{hae97}
J.~H{\"a}rterich.
 {Attractors of Viscous Balance Laws}.
 Dissertation, Freie {U}niversit\"at {B}erlin, 1997.
\bibitem[H{\"a}98]{hae98a}
J.~H{\"a}rterich.
 Attractors of viscous balance laws: Uniform estimates for the
  dimension.
 {\em J.~Diff.~Eqns.} {\bf 142} (1998), 188--211.
\bibitem[H{\"a}99]{hae99b}
J.~H{\"a}rterich.
 Equilibrium solutions of viscous scalar balance laws with a convex
  flux.
 {\em Nonlin.~Diff.~Eqns.~Appl.} {\bf 6} (1999), 413--436.
\bibitem[HaMi91]{hatmis91}
H.~Hattori and K.~Mischaikow.
 A dynamical system approach to a phase transition problem.
 {\em J.~Diff.~Eqns.} {\bf 94} (1991), 340--378.
\bibitem[He89]{Hei} S. Heinze. {\it Travelling waves for semilinear
parabolic partial differential equations in cylindrical domains.}
Dissertation, Heidelberg, 1989.
\bibitem[He81]{Henry}
D. Henry. {\em Geometric Theory of Semilinear Parabolic Equations}. 
Lect. Notes Math. {\bf 804}, Springer-Verlag, New York, Berlin,
Heidelberg, 1981.
\bibitem[He85]{henry85a}
D.~Henry.
 Some infinite dimensional {M}orse-{S}male systems defined by
  parabolic differential equations.
 {\em J. Diff. Eqns.} {\bf 59} (1985), 165--205.
\bibitem[HeWi91]{henwin91}
C.~Henze and A.~T. Winfree.
 A stable knotted singularity in an excitable medium.
 {\em Int. J. Bif. Chaos} {\bf 1} (1991), 891--922.
\bibitem[Hi83]{hirsch83}
M.~W. Hirsch.
 Differential equations and convergence almost everywhere in strongly
  monotone semiflows. {J}.~{S}moller, (ed.).
 In {\em Nonlinear Partial Differential Equations}. p. 267--285, AMS
  Publications, Providence, 1983.
\bibitem[Hi85]{hirsch85}
M.~W. Hirsch.
 Systems of differential equations that are competitive or cooperative
  {II}. {C}onvergence almost everywhere.
 {\em SIAM J. Math. Analysis} {\bf 16} (1985), 423--439.
\bibitem[Hi88]{hirsch88}
M.~W. Hirsch.
 Stability and convergence in strongly monotone dynamical systems.
 {\em Crelle J.~reine~angew.~Math.} {\bf 383} (1988), 1--58.
\bibitem[Hi\&al77]{hipush77}
M.~W. Hirsch, C.C. Pugh, and M.~Shub.
 {\em Invariant Manifolds}.
 Springer-Verlag, Berlin, 1977.
\bibitem[IoMi91]{IMD} 
G. Iooss and A. Mielke.
Bifurcating time--periodic solutions of
Navier--Stokes equations in infinite cylinders.
{\em  J. Nonlinear Science} {\bf 1} (1991), 107--146.
\bibitem[J{\"a}Lu92]{jaeluc92}
W.~J{\"a}ger and S.~Luckhaus.
 On explosions of solutions to a system of partial differential
  equations modelling chemotaxis.
 {\em Trans. Am. Math. Soc.}, {\bf 329}  (1992), 819--824.
\bibitem[Ja\&al88]{jahewi88}
W.~Jahnke, C.~Henze, and A.T. Winfree.
 {Chemical vortex dynamics in the 3-dimensional excitable media}.
 {\em Nature} {\bf 336} (1988), 662--665.
\bibitem[Ja\&al89]{Win}
W.~Jahnke, W.E. Skaggs, and A.T. Winfree.
Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and
in the two-variable Oregonator model.
{\em J. Chem. Phys}. {\bf 93} (1989), 740--749.
\bibitem[Ka66]{Kato}
T. Kato.
{\em Perturbation Theory for Linear Operators.}
Springer, Berlin, Heidelberg, New York, 1966. 
\bibitem[KaHa95]{kaha95} A. Katok and B. Hasselblatt.
{\em Introduction to the modern theory of dynamical systems.}
With a supplementary chapter by Katok and Leonardo
Mendoza. Encyclopedia of Mathematics and its Applications {\bf 54}, 
Cambridge University Press, Cambridge, 1995. 
\bibitem[KaHa02]{katok}
A.~Katok and B.~Hasselblatt (eds.) {\em {H}andbook of {D}ynamical
{S}ystems} {\bf 1}, 
{E}lsevier, {A}msterdam.
to appear 2002.
\bibitem[Ka87]{kauffm87}
L.H. Kauffman.
 {\em On Knots}.
 Princeton University Press, New Jersey, 1987.
\bibitem[Ke92]{keener92}
J.P. Keener. The core of the spiral.
 {\em SIAM J. Appl. Math.} {\bf 52} (1992), 1370--1390.
\bibitem[KeTy92]{keetys92}
J.P. Keener and J.J. Tyson.
 The dynamics of scroll waves in excitable media.
 {\em SIAM Rev.}, {\bf 34} (1992), 1--39.
\bibitem[Ki82]{Ki82} K. Kirchg\"assner.
 Wave-solutions of reversible systems and applications.
{\em J. Differential Equations} {\bf 45} (1982), 113--127.
\bibitem[Ki00]{kiyash00}
S.V. Kiyashko.
 The generation of stable waves in faraday experiment.
 {\em 2000 Int. Symp. Nonlinear Theory and its Applications}, 2000.
\bibitem[KoHo73]{KoHo} 
N.\,Kopell and L.N.\,Howard.
Plane wave solutions to reaction-diffusion equations.
{\it Studies in Appl. Math.} {\bf 52} (1973),  291--328.
\bibitem[KoHo81]{Kop}
N.\,Kopell and L.N.\,Howard.
Target patterns and spiral solutions to reaction-diffusion
equations with more than one space dimension. 
{\it Adv. Appl. Math.} {\bf 2} (1981), 417--449.
\bibitem[Ko02]{kozlov02}
V.V. Kozlov. {\em General Theory of Vortices}. Enc. Math. Sc. {\bf 67}, Dynamical Systems {X}. Springer-Verlag, New York, 2002.
\bibitem[Kr90]{krupa90}
M.~Krupa.
 {Bifurcations of relative equilibria}.
 {\em SIAM J. Math. Analysis} {\bf 21} (1990), 1453--1486.
\bibitem[KuMa83]{kubmar83}
M.~Kubi\v{c}ek and M.~Marek.
 {\em Computational Methods in Bifurcation Theory and Dissipative
  Structures}.
 Springer-Verlag, New York, 1983.
\bibitem[Ku95]{kuznet95}
Y.A. Kuznetsov.
 {\em Elements of Applied Bifurcation Theory}.
 Springer-Verlag, Berlin, 1995.
\bibitem[La91]{ladyzh91}
O.A. Ladyzhenskaya.
 {\em Attractors for Semigroups and Evolution Equations}.
 Cambridge University Press, 1991.
\bibitem[LaLi59]{Landau} L.D. Landau and E.M. Lifschitz. 
{\em Fluid Mechanics.}
Pergamon Press, London, 1959.
\bibitem[Li90]{lin90}
X.-B. Lin.
 Using {M}elnikov's method to solve {S}hilnikov's problems.
 {\em Proc. Roy. Soc. Edinburgh}, {\bf 116A} (1990), 295--325.
\bibitem[Ou\&al]{Ouy}
G. Li, Q.~Ouyang, V.~Petrov, and H.~L.~Swinney.
Transition from simple rotating chemical spirals to meandering
and traveling spirals.  
{\em Phys. Rev. Lett.} {\bf 77} (1996), 2105--2108.
\bibitem[Lo04]{love04}
A.E.H. Love.
 {\em A Treatise on the Mathematical Theory of Elasticity}.
 Dover Publications, New-York, 1904.
\bibitem[MP88]{mallet88}
J.~Mallet-Paret.
 Morse decompositions for delay-differential equations.
 {\em J.~Diff.~Eqns.} {\bf 72} (1988), 270--315.
\bibitem[MPSm90]{malsmi90}
J.~Mallet-Paret and H.~Smith.
 The {P}oincar\'e-{B}endixson theorem for monotone cyclic feedback
  systems.
 {\em J.~Diff.~Eqns.} {\bf 4} (1990), 367--421.
\bibitem[MPSe96a]{malsel96b}
J.~Mallet-Paret and G.R. Sell.
 The {P}oincar\'e-{B}endixson theorem for monotone cyclic feedback
  systems with delay.
 {\em J. Diff. Eqns.} {\bf 125} (1996), 441--489.
\bibitem[MPSe96b]{malsel96a}
J.~Mallet-Paret and G.R. Sell.
 Systems of differential delay equations: Floquet multipliers and
  discrete {L}yapunov functions.
 {\em J. Diff. Eqns.} {\bf 125} (1996), 385--440.
\bibitem[MaPa97]{Panfilov} 
A.F.M. Maree and A.V. Panfilov.
Spiral breakup in excitable tissue due to lateral instability.
{\em Phys. Rev. Lett.} {\bf 78} (1997), 1819--1822.
\bibitem[Ma78]{matano78}
H.~Matano.
 Convergence of solutions of one-dimensional semilinear parabolic
  equations.
 {\em J. Math. Kyoto Univ.}, {\bf 18} (1978), 221--227.
\bibitem[Ma79]{matano79}
H.~Matano.
 Asymptotic behavior and stability of solutions of semilinear
  diffusion equations.
 {\em Publ.~Res.~Inst.~Math.~Sc.~Kyoto~Univ.} {\bf 15}
  (1979), 401--454.
\bibitem[Ma82]{matano82}
H.~Matano.
Nonincrease of the lap-number of a solution for a one-dimensional
  semi-linear parabolic equation.
 {\em J. Fac. Sci. Univ. Tokyo Sec. IA}, {\bf 29}(1982), 401--441.
\bibitem[Ma86]{matano86b}
H.~Matano.
 Strongly order-preserving local semi-dynamical systems --- theory and
  applications.
 In {\em Semigroups, Theory and Applications}. H. Brezis, M.G.
  Crandall, F. Kappel (eds.), 178--189. John Wiley \& Sons, New York, 1986.
\bibitem[Ma87]{matano87}
H.~Matano.
 Strong comparison principle in nonlinear parabolic equations.
 In {\em Nonlinear Parabolic Equations: Qualitative Properties of Solutions}, L. Bocardo, A. Tesei (eds.), 148--155.
 Pitman Res. Notes Math. Ser. {\bf 149} (1987).
\bibitem[Ma88]{matano88}
H.~Matano.
 Asymptotic behavior of solutions of semilinear heat equations on ${S^1}$.
 In {\em Nonlinear Diffusion Equations and their Equilibrium States
  {II} }. W.-M. Ni, L.A. Peletier, J. Serrin (eds.). 139--162. Springer-Verlag,
  New York, 1988.
\bibitem[MaNa97]{matnak97}
H.~Matano and K.-I. Nakamura.
The global attractor of semilinear parabolic equations on ${S^1}$.
{\em Discr. Contin. Dyn. Syst.} {\bf 3 } (1997), 1--24.
\bibitem[MaWi89]{mawwil89}
J.~Mawhin and M.~Willem.
 {\em Critical Point Theory and Hamiltonian Systems}.
 Springer-Verlag, New York, 1989.
\bibitem[Mi91]{mielke91}
A.~Mielke.
 {\em Hamiltonian and Lagrangian Flows on Center Manifolds}.
 Springer-Verlag, Berlin, 1991.
\bibitem[Mi94]{mielke94}
A.~Mielke.
 Essential manifolds for elliptic problems in infinite cylinders.
 {\em J. Diff. Eqns.}, {\bf 110} (1994), 322--355.
\bibitem[Mi97]{MielkeSH} A. Mielke.
Instability and stability of rolls in the Swift-Hohenberg equation.
{\em Comm. Math. Phys.} {\bf 189} (1997), 829--853.
\bibitem[MiSc96]{SchMie} A. Mielke and G. Schneider.  Derivation and
justification of the complex Ginzburg-Landau equation as a modulation
equation.  Dynamical systems and probabilistic methods in partial
differential equations (Berkeley, CA, 1994), 191--216, {\em  Lectures in
Appl. Math.} {\bf 31}, Amer. Math. Soc., Providence, RI, 1996.
\bibitem[MiZy91]{Mikhailov} A.S. Mikhailov and V.S. Zykov. 
Kinematical theory of spiral waves in excitable media: comparison
with numerical simulations.
{\em Physica D} {\bf 52} (1991), 379--397.
\bibitem[Mo73]{moser73}
J.~Moser.
 {\em Stable and Random Motions in Dynamical Systems}.
 Princeton University Press, New York, 1973.
\bibitem[M{\"u}Zy94]{muezyk94}
S.C. M{\"u}ller and V.S. Zykov.
 Simple and complex spiral wave dynamics.
 {\em Phil. Trans. Roy. Soc. Lond. A} {\bf 347} (1994), 677--685.
\bibitem[Na90]{nadira90}
N.S. Nadirashvili.
 On the dynamics of nonlinear parabolic equations.
 {\em Soviet Math. Dokl.} {\bf 40} (1990), 636--639.
\bibitem[Ne\&al]{ertl93}
S.~Nettesheim, A.~von Oertzen, H.H. Rotermund, and G.~Ertl.
Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110)
  --- front propagation and spiral waves.
{\em J. Chem. Phys.} {\bf 98} (1993), 9977--9985.
\bibitem[OgNa00]{oginak00}
T.~Ogiwara and K.-I. Nakamura.
 Spiral traveling wave solutions of some parabolic equations on
  annuli.
 {\em {I}n Nonlinear Analysis, Josai Math. Monogr., Nishizawa, Kiyoko
  (ed.)} {\bf 2} (2000), 15--34.
\bibitem[Ol02]{oliva02}
W.M. Oliva.
 Morse-{S}male semiflows. {O}peness and {A}-stability.
 {\em Comm. Fields Inst.} (2002), in press.
\bibitem[Pa61]{palais61}
R.~S. Palais.
 On the existence of slices for actions of non-compact {L}ie groups.
 {\em Ann. of Math.} {\bf 73} (1961), 295--323.
\bibitem[Pa69]{palis69}
J.~Palis.
 On {M}orse-{S}male dynamical systems.
 {\em Topology} {\bf 8} (1969), 385--404.
\bibitem[PaSm70]{palsma70}
J.~Palis and S.~Smale.
 Structural stability theorems.
 In {\em Global Analysis}. Proc. Symp. in Pure Math.~vol.~XIV.~AMS,
  Providence, 1970.
 S. Chern, S. Smale (eds.).
\bibitem[Pa88]{Palmer} K.J. Palmer.
Exponential dichotomies and Fredholm operators.
{\em Proc. Amer. Math. Soc.} {\bf 104} (1988), 149--156.
\bibitem[PaWi85]{panwin85}
A.V. Panfilov and A.~T. Winfree.
 Dynamical simulations of twisted scroll rings in 3-dimensional
  excitable media.
 {\em Physica D} {\bf 17} (1985), 323--330.
\bibitem[Pa83]{pazy83}
A.~Pazy.
 {\em Semigroups of Linear Operators and Applications to Partial
  Differential Equations}.
 Springer-Verlag, New York, 1983.
\bibitem[Pe\&al91]{PMExp}
V.~Perez-Munuzuri, R.~Aliev, B.~Vasiev, V.~Perez-Villar, and V.~I.~Krinsky.
Super-spiral structure in an excitable medium.
{\em Nature} {\bf 353} (1991) 740--742.
\bibitem[Pe\&al93]{PMtheory} V.~Perez-Munuzuri, M.~Gomez-Gesteira, and
V.~Perez-Villar.  A geometrical-kinematical approach to spiral
wave formation: Super-spiral waves.  
{\em Physica~D} {\bf 64} (1993),
420--430.
\bibitem[Pe\&al97]{PSS} D. Peterhof, A. Scheel, and B. Sandstede. 
Exponential dichotomies for solitary wave solutions of semilinear
elliptic equations on infinite cylinders. {\it J. Diff. Eqns.} {\bf 140}
(1997),  266--308.
\bibitem[PlBo96]{plabod96}
B.B. Plapp and E.~Bodenschatz.
 Core dynamics of multiarmed spirals in {R}ayleigh-{B}\'enard
  convection.
 {\em Physica Scripta} {\bf 67} (1996), 111--117.
\bibitem[Po89]{polaci90}
P.~Pol\'a\v{c}ik.
 Convergence in strongly monotone flows defined by semilinear
  parabolic equations. {\em J. Diff. Eqs.} {\bf 79} (1989), 89--110.
\bibitem[Po95]{polaci95}
P.~Pol\'a\v{c}ik.
 High-dimensional $\omega$-limit sets and chaos in scalar parabolic
  equations.
 {\em J. Diff. Eqns.}, {\bf 119} (1995), 24--53.
\bibitem[Po02]{polaci02}
P.~Pol\'a\v{c}ik.
 Parabolic equations: Asymptotic behavior and dynamics on invariant
  manifolds.
 In {\em Handbook of Dynamical Systems, Vol.} {\bf 2}. B. Fiedler (ed.),
  Elsevier, Amsterdam, 2002.
 In press.
\bibitem[Po33]{polya} G.~Polya.
Qualitatives {\"u}ber {W}{\"a}rmeaustausch. 
{\em Z. {A}ngew. Math. Mech.} {\bf 13} (1933), 125--128,.
\bibitem[Po92]{pospie92}
G.~Pospiech.
 {\em {Eigenschaften, Existenz und Stabilit{\"a}t von travelling wave
  L{\"o}sungen zu einem System von Reaktions-Diffusions-Gleichungen}}.
 {Dissertation}, Universit{\"a}t Heidelberg, 1992.
\bibitem[PrRy98a]{priryb98a}
M.~Prizzi and K.P. Rybakowski.
 Complicated dynamics of parabolic equations with simple gradient
  dependence.
 {\em Trans. Am. Math. Soc.} {\bf 350} (1998), 3119--3130.
\bibitem[PrRy98b]{priryb98b}
M.~Prizzi and K.P. Rybakowski.
 Inverse problems and chaotic dynamics of parabolic equations on
  arbitrary spatial domains.
 {\em J. Diff. Eqns.} {\bf 142} (1998), 17--53.
\bibitem[PrWe67]{prowei67}
M.H. Protter and H.F. Weinberger.
 {\em Maximum Principles in Differential Equations}.
 Prentice Hall, Englewood Cliffs, New Jersey, 1967.
\bibitem[Ra02]{raugel02}
G.~Raugel.
Global attractors.
In {\em Handbook of Dynamical Systems, Vol. 2}. B. Fiedler (ed.),
Elsevier, Amsterdam, 2002.
In press.
\bibitem[ReSi78]{blochwave} M. Reed and B. Simon. 
{\em Methods of Modern Mathematical Physics IV.}
Academic Press, 1978.
\bibitem[RoSa95]{RS95}
J. Robbin and D. Salamon. The spectral flow and the Maslov
index.  {\em Bull. London Math. Soc}. {\bf 27} (1995), 1--33.
\bibitem[Ro85]{ro85} C. Rocha.
Generic properties of equilibria of reaction-diffusion equations with
variable diffusion.{\em  Proc. R. Soc. Edinb.} A {\bf 101} (1985), 45--55.
\bibitem[Ro91]{ro91} C. Rocha.
Properties of the attractor of a scalar parabolic PDE.
{\em J. Dyn. Differ. Equations} {\bf 3} (1991), 575--591. 
\bibitem[Sa93a]{sandst93a}
B.~Sandstede.
 {\em Verzweigungstheorie homokliner {V}erdopplungen.}
 Dissertation, Universit{\"a}t Stuttgart, 1993.
\bibitem[Sa93b]{sandst93b}
B.~Sandstede.
 Asymptotic behavior of solutions of non-autonomous scalar
  reaction-diffusion equations.
 {\em In Conf. Proceeding International Conference on Differential
  Equations, Barcelona 1991, C. Perello, C. Simo, and J. Sola-Morales (eds.),
  888--892, World Scientific, Singapore}, 1993.
\bibitem[SaFi92]{sanfie92}
B.~Sandstede and B.~Fiedler.
 Dynamics of periodically forced parabolic equations on the circle.
 {\em Ergod. Theor. Dynam. Sys.} {\bf 12} (1992), 559--571.
\bibitem[SaSc99]{SSmodpulse} B. Sandstede and A. Scheel.  Essential
instability of pulses and bifurcations to modulated travelling waves. 
 {\it Proc. Roy. Soc. Edinburgh. A} {\bf 129} (1999), 1263--1290.
\bibitem[SaSc00a]{SSgluing} B. Sandstede and A. Scheel.Gluing unstable fronts 
and backs together can produce stable pulses. {\it  Nonlinearity} {\bf 13}
(2000), 1465--1482.
\bibitem[SaSc00b]{SSmodstab}  B. Sandstede and A. Scheel. Spectral
stability of modulated travelling waves bifurcating near essential
instabilities. {\it Proc. R. Soc. Edinburgh A} {\bf 130} (2000), 419--448.  
\bibitem[SaSc00c]{SStrunc} B. Sandstede and A. Scheel. Absolute and
convective instabilities of waves on unbounded and large bounded
domains.  {\it Physica D} {\bf 145} (2000), 233--277. 
\bibitem[SaSc00d]{SSspiralstability} B. Sandstede and A. Scheel. Absolute 
versus convective instability of spiral waves.  {\it  Phys. Rev. E}. {\bf 62}
 (2000), 7708--7714.
\bibitem[SaSc01a]{SSsuperspiral} B. Sandstede and A. Scheel. Super-spiral
structures of meandering and drifting spiral waves.
 {\it Phys. Rev. Lett}. {\bf 86} (2001), 171--174. 
\bibitem[SaSc01b]{SSfront} B. Sandstede and A. Scheel.  Essential
instabilities of fronts: bifurcation and bifurcation failure.
{\it Dynamical Systems: An International Journal} \textbf{16} (2001), 1--28.
\bibitem[SaSc01c]{SSperiodic} B. Sandstede and A. Scheel. On the stability
of periodic travelling waves with large spatial period. 
  {\it J. Diff. Eqns.}  {\bf 172} (2001), 134--188.
\bibitem[SaSc01d]{SSmathnach} B. Sandstede and A. Scheel.  On the
structure of spectra of modulated travelling waves.{\it  Math. Nachr.} {\bf
232} (2001), 39--93.
\bibitem[SaSc02a]{SSconv}  B. Sandstede and A. Scheel.  Nonlinear
convective stability and instability --- the role of absolute
spectra and nonlinearities.{\em In preparation} (2002).
\bibitem[SaSc02b]{SSmea}  B. Sandstede and A. Scheel. Instabilities of
spiral waves in large disks.{\em In preparation} (2002).
\bibitem[Sa\&al97a]{SSW} B. Sandstede, A. Scheel, and C. Wulff.
Center manifold reduction for spiral wave dynamics.
 {\it C. R. Acad. Sci. Paris, S\'erie I} {\bf 324} (1997), 153--158.
\bibitem[Sa\&al97b]{SSW1} B. Sandstede, A. Scheel, and C. Wulff. 
Dynamics of spiral waves on unbounded domains using center-manifold
reduction. 
{\it J. Diff. Eqns.} {\bf 141} (1997),  122--149.
\bibitem[Sa\&al99]{SSW2} B. Sandstede, A. Scheel, and C. Wulff.
Bifurcations and dynamics of spiral waves. 
 {\it J. Nonlinear Science} {\bf 9} (1999), 439--478.
\bibitem[Sc90]{schaaf90}
R.~Schaaf.
 {\em Global Solution Branches of Two Point Boundary Value Problems}.
 Springer-Verlag, New York, 1990.
\bibitem[Sc96]{ScheelDiss} A. Scheel.  
Existence of fast travelling waves for some parabolic equations
--- a dynamical systems  approach.
{\it J. Dyn. Diff. Eqns}. {\bf 8} (1996), 469--548.
\bibitem[Sc97]{Sspiralsub} A. Scheel. Subcritical bifurcation to
infinitely many rotating waves.  
J. Math. Anal. Appl. {\bf 215} (1997), 252--261. 
\bibitem[Sc98]{Sspiralex} A. Scheel. Bifurcation to spiral waves in
reaction-diffusion systems.  {\it  SIAM J. Math. Anal}. {\bf 29} (1998),
1399--1418.
\bibitem[Sc01]{Shabil} A. Scheel.Radially symmetric patterns of
reaction-diffusion systems. {\em Preprint} 2001.
\bibitem[Sc98b]{guidohopf} G. Schneider.
Hopf bifurcation in spatially extended reaction-diffusion systems.
{\em J. Nonlinear Sci}. {\bf 8} (1998), 17--41.
\bibitem[Sc98c]{Schneider} G. Schneider.
Nonlinear diffusive stability of spatially periodic
solutions --- abstract theorem and higher space dimensions.
{\em Tohoku Math. Publ.} {\bf 8} (1998), 159--167.
\bibitem[Si89]{sinai89}
Ya.G. Sinai.
 {\em Ergodic theory with applications to dynamical systems and
  statistical mechanics}.
 Enc. Math. Sc. {\bf  2}, Dynamical Systems {II}. Springer-Verlag, Berlin,
  1989.
\bibitem[SkSw91]{SkSw}
G.~S.~Skinner and H.~L.~Swinney. 
Periodic to quasiperiodic
transition of chemical spiral rotation.
{\em Physica~D}  {\bf 48} (1991), 1--16.
\bibitem[Sm95]{smith95}
H.~Smith.
 {\em Monotone Dynamical Systems: An Introduction to the Theory of
  Competitive and Cooperative Systems}.
 AMS, Providence, 1995.
\bibitem[Sm83]{smolle83}
J.~Smoller.
 {\em Shock Waves and Reaction-Diffusion Equations}.
 Springer-Verlag, New York, 1983.
\bibitem[St00]{steven00}
A.~Steven.
 The derivation of chemotaxis equations as limit dynamics of

  moderately interacting stochastic many-particle systems.
 {\em SIAM J. Appl. Math.}, {\bf 61} (2000), 183--212.
\bibitem[St90]{struwe90}
M.~Struwe.
 {\em Variational Methods}.
 Springer-Verlag, Berlin, 1990.
\bibitem[St36]{stu}
C.~Sturm.
Sur une classe d'\'equations \`a diff\'erences partielles.
{\em J.~Math.~Pure~Appl.} {\bf 1} (1836), 373--444,.
\bibitem[Ta74]{takens74}
F.~Takens.
 Singularities of vector fields.
 {\em Publ.~IHES}, {\bf 43} (1974), 47--100.
\bibitem[Ta79]{tanabe79}
H.~Tanabe.
 {\em Equations of Evolution}.
 Pitman, Boston, 1979.
\bibitem[Te88]{temam88}
R.~Temam.
 {\em Infinite-Dimensional Dynamical Systems in Mechanics and
  Physics}.
 Springer-Verlag, New York, 1988.
\bibitem[ToKn98]{TK}
S.M. Tobias and E. Knobloch.
Breakup of spiral waves into chemical turbulence.
{\em Phys. Rev. Lett.} {\bf 80} (1998), 4811--4814.
\bibitem[Tu52]{Tur} A. Turing.
The chemical basis of morphogenesis.
{\em Phil. Trans. Roy. Soc. B} {\bf 237} (1952), 37--72.
\bibitem[TySt91]{tysstr91}
J.J. Tyson and S.H. Strogatz.
 The differential geometry of scroll waves.
 {\em Int. J. Bif. Chaos}, {\bf 1} (1991), 723--744.
\bibitem[Uh76]{uhlenb76}
K.~Uhlenbeck.
 Generic properties of eigenfunctions.
 {\em Amer. J. Math.}, {\bf 98} (1976), 1059--1078.
\bibitem[Un\&al93]{muelle93}
Zs. Ungvarai-Nagy, J. Ungvarai, and S.C. M{\"u}ller.
Complexity in spiral wave dynamics.
{\em Chaos} {\bf 3} (1993), 15--19.
\bibitem[Va82]{vander82}
A.~Vanderbauwhede.
 {\em Local Bifurcation and Symmetry}.
 Pitman, Boston, 1982.
\bibitem[Va89]{van} A. Vanderbauwhede.
Center manifolds, normal forms and elementary bifurcations.
{\it  Dynamics Reported} {\bf 2} (1989), 89--169.
\bibitem[Wa70]{walter70}
W.~Walter.
 {\em Differential and Integral Inequalities}.
 Springer-Verlag, New York, 1970.
\bibitem[WiRo46]{wieros46}
N.~Wiener and A.~Rosenblueth.
The mathematical formulation of the problem of conduction of impulses
  in a network of connected excitable elements, specifically in
cardiac muscle.
 {\em Arch. Inst. Cardiol. Mexico} {\bf 16} (1946), 205--265.
\bibitem[Wi72]{winfre72}
A.~T. Winfree.
Spiral waves of chemical activity.
{\em Science}, {\bf 175} (1972), 634--636.
\bibitem[Wi73]{winfre73}
A.~T. Winfree.
Scroll-shaped waves of chemical activity in three dimensions.
{\em  Science} {\bf 181} (1973), 937--939.
\bibitem[Wi87]{winfre87}
A.~T. Winfree.
{\em When Time Breaks Down}.
Princeton University Press, Princeton, NJ, 1987.
\bibitem[Wi91]{winfre91}
A.~T. Winfree.
Varieties of spiral wave behavior: An experimentalist's approach to
the theory of excitable media.
{\em Chaos} {\bf 1} (1991), 303--334.
\bibitem[Wi95]{winfre95}
A.~T. Winfree.
 Persistent tangles of vortex rings in excitable media.
 {\em Physica D} {\bf 84} (1995), 126--147.
\bibitem[Wi01]{winfre80}
A.~T. Winfree. The geometry of biological time. {\em Biomathematics}
{\bf 8}, Springer-Verlag, Berlin-New York, 2001.
\bibitem[Wi\&al95]{wiwise85}
A.~T. Winfree, E.M. Winfree, and M.~Seifert.
 Organizing centers in a cellular excitable medium.
 {\em Physica D}, {\bf 17} (1995), 109--115.
\bibitem[Wo98]{woldis}Matthias Wolfrum.
{\em Geometry of Heteroclinic Cascades in Scalar Semilinear Parabolic Equations.} 
Dissertation, Freie Universit\"at Berlin, 1998.
\bibitem[Wo02a]{wolper} M. Wolfrum. Personal communication, (2002).
\bibitem[Wo02b]{wolfconstr} M. Wolfrum. A sequence of order relations,
encoding heteroclinic connections in scalar parabolic PDEs. J. Diff. Eqns., to appear (2002).
\bibitem[Wu96]{wulff96} C.~Wulff. {\em {Theory of Meandering and
Drifting Spiral Waves in Reaction-Diffusion Systems}}.
{Dissertation}, Berlin, 1996.
\bibitem[Wu\&al01]{wulame01} C. Wulff, J. Lamb, and I. Melbourne.
Bifurcation from relative periodic solutions. 
{\em  Ergodic Theory Dynam. Systems} {\bf 21} (2001), 605--635.
\bibitem[Ya\&al98]{yamiya98}
H.~Yagisita, M.~Mimura, and M.~Yamada.
 Spiral wave behaviors in an excitable reaction-diffusion system on a
  sphere.
 {\em Physica D} {\bf 124} (1998), 126--136.
\bibitem[Ze85]{zeidle85}
E.~Zeidler.
 {\em Nonlinear functional analysis and its applications. {III}:
  Variational methods and optimization}.
 Springer-Verlag, New York, 1985.
\bibitem[Ze93]{zeidle93}
E.~Zeidler.
 {\em Nonlinear functional analysis and its applications. Volume I:
  Fixed-point theorems}.
 Springer-Verlag, New York, 1993.
\bibitem[Ze68]{zeleny68}
T.I. Zelenyak.
Stabilization of solutions of boundary value problems for a second
order parabolic equation with one space variable.
{\em Diff.~Eqns.} {\bf 4} (1968), 17--22.
\bibitem[ZhOu00]{ZO}
L.~Q.~Zhou and Q.~Ouyang.
Experimental studies on long-wavelength instability and spiral breakup in a reaction-diffusion system.
{\em Phys. Rev. Lett.} {\bf 85} (2000), 1650--1653.

\end{thebibliography}
\end{document}



%\newpage
%\bibliography{References}


%{matfin98}   "V.M. Matrosov and I.A. Finogenko
%{matros71    "V.M. Matrosov",%

%scheel98     "A.~Scheel
%sansch99     "B. Sandstede and  A. Scheel",
%sansch00 	   "B. Sandstede and  A. Scheel"
%sansch00a    "B. Sandstede and  A. Scheel",
%frbosc99     "G. Franceschini and S. Bose and E. Sch"oll"
%fieman98     "B.~Fiedler and R.-M. Mantel",



%





