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Volume 5, Issue 4, 2008, pp. 3201–3271
DOI: 10.4171/OWR/2008/57

Published online: 2009-09-30

Dynamics of Patterns

Wolf-Jürgen Beyn[1], Bernold Fiedler[2] and Björn Sandstede[3]

(1) Universität Bielefeld, Germany
(2) Freie Universität Berlin, Germany
(3) Brown University, Providence, United States

Ideas from dynamical systems have had a profound impact on the way we think about pattern formation. Bifurcation theory, for instance, has helped tremendously in explaining pattern selection in experiments, including Rayleigh--Benard convection and Belousov--Zhabotinsky reactions. However, these results can typically only describe patterns with a given prescribed periodic lattice structure on the plane. Amplitude equations go beyond this limitation: They allow us to investigate the dynamics of slowly varying amplitude modulations of a fixed spatially homogeneous state over large, but finite, time intervals.

Over the past few years, the focus has shifted to situations where neither bifurcation theory nor the amplitude-equation formalism can give enough insight into the formation and the dynamics of patterns. Examples are the dynamical selection of patterns, extracting and describing transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects and behaviours. This workshop brought together researchers who work on these questions from different perspectives and with different techniques, ranging from dynamical systems theory, qualitative analysis of partial differential equations, and bifurcation theory to spectral analysis and numerical methods for patterns.

During the workshop, 25 presentations were given. In addition, three PhD students discussed their projects in shorter talks of 15 minutes length. On Tuesday afternoon, no talks were scheduled. Instead, the attendees had the opportunity to discuss more specialized topics in smaller groups. We now describe briefly the main outcomes and new directions that emerged during the workshop.

The formation and interaction of pulses in one space dimension were one central theme of the workshop. Recent efforts aim to describe the interaction of localized pulses analytically and to compute interacting pulses efficiently, using numerical means. Progress was made in particular for pulses that are only weakly localized: in certain circumstances, it is then still possible to capture the interaction of such pulses analytically.

Over the past few years, the \emph{freezing method} has been investigated thoroughly from both analytical and numerical viewpoints. This method computes pulses numerically by separating the shape dynamics from the dynamics on the underlying symmetry group. These developments were discussed together with applications to spiral waves and to propagating pulses in partial differential equations of mixed type.

With all these successes, it became clear, however, that both analytic and numerical understanding of the evolution and interaction of two-dimensional localized spatio-temporal patterns is still very rudimentary.

Spatially extended patterns and their dynamics constituted a further focus. Much recent work has centered on explaining specific phenomena that have been observed experimentally: examples include turbulent stripe patterns in fluid flows, Liesegang precipitation patterns, planar hexagon patches, and vortex dynamics in flows past cylinders. Significant progress was also made in proving spectral and nonlinear stability of spatially extended waves such as rotating waves, spatially homogeneous oscillations, and spatially periodic structures. Furthermore, techniques to assess spectral stability for multi-dimensional fronts were discussed.

Systems with delay form an important class of infinite-dimensional systems that exhibit interesting dynamical patterns. State-dependent delays allow the delay of the system to depend on the prehistory state of that system itself. Hysteresis is one well-known example. State-dependent delays are motivated by important applications, generate a plethora of new dynamical patterns, and present formidable obstacles to analysis. Progress reports included patterns of periodicity in hysteresis, implicitly defined delays, and singularly perturbed equations.

\subsection*{Special Tuesday sessions}

No talks were scheduled on Tuesday afternoon to give participants an additional opportunity for discussion in smaller groups. We briefly report on two group meetings that took place in this setting.

\subsubsection*{Poster discussion: More on pulses, shocks, and their interactions.}

The main intention of the posters during the work session was to discuss among a group of specialists the existence, stability and bifurcation of nonlinear waves that are either periodic in time or in space. The group discussed progress to these kinds of problems that involved various different approaches such as singular perturbation techniques, modulation equations, and pointwise Green's function estimates. Specific topics and the corresponding contributors were: \begin{itemize} \item Busse balloons and bifurcations of spatially periodic patterns (Arjen Doelman, with Harmen von der Ploeg, Jens Rademacher and Sjors van der Stelt); \item Interfaces between rolls in the Swift--Hohenberg equation (Mariana Haragus with Arnd Scheel); \item Delayed bifurcation in a simple reaction-diffusion equation (Tasso Kaper with Peter De Maesschalck and Nikola Popovic); \item Stability of time-periodic viscous shocks (Bj\"orn Sandstede with Margaret Beck and Kevin Zumbrun). \end{itemize} Another intention of the poster discussion was to have interactions between this group and junior researchers who were given the possibility to present more details (in particular numerical results) than during their short talks: \begin{itemize} \item Freezing waves in hyperbolic PDEs (Jens Rottmann-Matthes); \item Numerical decomposition of multistructures (Sabrina Selle). \end{itemize}

\subsubsection*{Global parabolic dynamics.} Progress and discussion addressed two aspects of the global dynamics of semilinear parabolic partial differential equations, mainly on a circle domain. These aspects are the Morse--Smale or Kupka--Smale property, on the one hand, and the characterization of global attractors, on the other hand. The Kupka--Smale property asserts hyperbolicity of all equilibria and periodic orbits, as well as transversality of their stable and unstable manifolds to hold for generic (i.e., for ``most'') nonlinearities. The Morse--Smale property asserts, in addition, the absence of any recurrence beyond periodicity. In such situations, it is conceivable, but still a formidable task, to study the detailed spatio-temporal structure of the patterns arising in the global attractor.

More precisely, we considered the following reaction-diffusion equations on the circle $S^1$: $$ u_t(x,t) = u_{xx}(x,t) + f(x, u(x,t), u_x(x,t)), \quad (x,t) \in S^1 \times {\bf R}^+, $$ where $f$ is a regular function from $S^1 \times {\bf R}^2$ into ${\bf R}$. First, we have recalled that these equations satisfy the Poincar\'{e}--Bendixson property [2]. We have also stated the recent result of Czaja and Rocha [1], who showed that the stable and unstable manifolds of hyperbolic periodic orbits intersect transversally.

Genevi{\`e}ve Raugel presented a proof of the Morse--Smale property for the above equation. She first showed genericity (with respect to the non-linearity) of the hyperbolicity of all equilibria and periodic orbits [3]. The main ingredients are the non-increase of the zero number and Sard--Smale theorems. She also showed automatic transversality of the stable and unstable manifolds of equilibria with different Morse indices, the generic non-existence (with respect to the non-linearity) of orbits connecting two equilibria with the same Morse index, etc. [4]. These properties allow to show that, generically with respect to the non-linearity $f$, the above reaction-diffusion equation on the circle $S^1$ indeed possesses the Morse--Smale property.

In the second part of this discussion, we have considered genericity (with respect to the non-linearity) of hyperbolicity of all equilibria and periodic orbits in the case of a scalar reaction-diffusion equation in higher dimension and explained the results already obtained in a work in progress by P. Brunovsk{\`y}, R. Joly, and G. Raugel. Genevi{\`e}ve Raugel noticed that the same types of arguments and techniques should lead them to the proof of the Kupka--Smale property for parabolic PDEs in the near future.

For $x$-independent nonlinearities $f=f(u,u_x)$ on the circle domain, the Morse--Smale property enters the description of all generic global attractors given in [5]. Carlos Rocha indicated how to characterize the set of $2\pi$-periodic solutions of planar Hamiltonians of the form $u'' + g(u) = 0$ and obtained a useful tool for the description of the associated global attractors. He discussed a permutation characterization for the periodic solutions of the corresponding stationary problems. Essentially, the permutation describes the braid formed by the stationary solutions and traveling waves of the semilinear parabolic equation [6]. Extending this result to equilibria in the $x$-reversible case $f(u,-p)=f(u,p)$, this characterization indeed extends to describe the precise heteroclinic structure of the above parabolic partial differential equation in this case.

{\bf References\medskip }

\noindent [1] R. Czaja and C. Rocha, {\it Transversality in scalar reaction-diffusion equations on a circle}, J. Diff. Eqs. {\bf 245} (2008), 692--721

\noindent [2] B. Fiedler and J. Mallet-Paret, {\it A Poincar\'{e}--Bendixson theorem for scalar reaction-diffusion equations}, Arch. Rational Mech. Analysis {\bf 107} (1989), 325--345.

\noindent [3] R. Joly and G. Raugel, {\it Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle}, to appear in Transactions of the AMS.

\noindent [4] R. Joly and G. Raugel, {\it Generic Morse--Smale property for the parabolic equation on the circle}, manuscript.

\noindent [5] B. Fiedler, C. Rocha, M. Wolfrum, {\it Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle}, J. Diff. Eqs. {\bf 201} (2004), 99--138.

\noindent [6] B. Fiedler, C. Rocha, M. Wolfrum, {\it A permutation characterization of periodic orbits in autonomous Hamiltonians or reversible systems}, Preprint (2009).

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Beyn Wolf-Jürgen, Fiedler Bernold, Sandstede Björn: Dynamics of Patterns. Oberwolfach Rep. 5 (2008), 3201-3271. doi: 10.4171/OWR/2008/57