Spatio-Temporal Dynamics of Reaction-Diffusion Patterns

Abstract

In this survey we look at parabolic partial differential equations from a dynamical systems point of view. With origins deeply rooted in celestial mechanics, and many modern aspects traceable to the monumental influence of Poincaré, dynamical systems theory is mainly concerned with the global time evolution T (t)u 0 of points u 0 -and of sets of such points -in a more or less abstract phase space X. The success of dynamical concepts such as gradient flows, invariant manifolds, ergodicity, shift dynamics, etc. during the past century has been enormous -both as measured by achievement, and by vitality in terms of newly emerging questions and long-standing open problems.In parallel to this development, the applied horizon now reaches far beyond the classical sources of celestial and Hamiltonian mechanics. Applications areas today include physics, many branches of engineering, economy models, and mathematical biology, to name just a few. This influence can certainly be felt in several articles of this volume and cannot possibly be adequately summarized in our survey.Some resources on recent activities in the area of dynamics are the book series Dynamical Systems I -X of the Encyclopedia of Mathematical Sciences [AnAr88, Si89, Ar&al88, ArNo90, Ar94a, Ar93, ArNo94, Ar94b, An91, Ko02], the Handbook of Dynamical System [BrTa02, Fi02, KaHa02], the Proceedings [Fi&al00], and the fundamental books [ChHa82,GuHo83,Mo73,KaHa95].In the context of partial differential equations, the phase space X of solutions u = u(t, x) becomes infinite-dimensional: typically a Sobolev space of spatial profiles u(t) = u(t, ·). More specifically, the evolution of u(t) = T (t)u 0 ∈ X with time t is complemented by the behavior of the x-profiles x → u(t, x) of solutions u. Such spatial profiles could be monotone or oscillatory; in dim x = 1 they could define sharp fronts or peaks moving at constant or variable speeds, with possible collision or mutual repulsion. Target pat- 22Bernold Fiedler and Arnd Scheel terns or spirals can emerge in dim x = 2. Stacks of spirals, which Winfree called scroll waves, are possible in dim x = 3. For a first cursory orientation in such phenomena and their mathematical treatment we again refer to [Fi02] and the article of Fife in the present volume, as well as the many references there.For our present survey, we focus on the spatio-temporal dynamics of the following rather "simple", prototypical parabolic partial differential equation:(2.1)Here t ≥ 0 denotes time, x ∈ Ω ⊆ R m is space and u = u(t, x) ∈ R N is the solution vector. We consider C 1 -nonlinearities f and constant, positive, diagonal diffusion matrices D. This eliminates the beautiful pattern formation processes due to chemotaxis; see [JäLu92, St00], for example. Note that we will not always restrict f to be a pure reaction term, like f = f (u) or f = f (x, u). More general than that, we sometimes allow for a dependence of f on ∇u to include advection effects. The domain Ω will be assumed smooth and bounded, typically with Dirichlet or Neumann boundary c...

Spatial Wavenumber Selection in Recurrent Precipitation

“…In this section, we therefore make precise this notion and show how to adapt the above dimension counting arguments. We refer to [22,20,28,12,29,30] for background on this type of equation.…”

Spatial decay of rotating waves in reaction diffusion systems

“…The difficulty is related to the fact that the essential spectrum touches the imaginary axis at infinitely many points. The spectrum of the linearization at (nonlocalized) spiral waves is well-known and has been extensively studied by Sandstede, Scheel and Fiedler in [16,31,32]. For numerical computations it is essential to truncate the equations (1.1), (1.3) and (1.9) to a bounded domain, so that standard approximations, e.g.…”

Coherent structures near the boundary between excitable and oscillatory media

“…We will then compute the location of point spectrum in a neighborhood of the origin. The point spectrum is actually part of the extended point spectrum and as such independent of weights, as long as Fredholm borders do not cross the location of the eigenvalue; see [13,29]. Accordingly, we will in the following refer to the point spectrum as extended point spectrum, emphasizing that it is in fact largely independent of the choice of function space.…”

“…Throughout we refer to [32,13] for definitions and results on the essential spectrum. We will refer to u ± (µ) as the equilibria, so that u + is unstable for the kinetics and u − is stable for the kinetics.…”

“…The Fredholm index of the linearization at a front in L 2 η changes whenever d ± (λ, −η + ik) = 0 for some k ∈ R [32,13]. The zero set defines algebraic curves in the complex plane which we refer to as Fredholm borders.…”