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