Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

Fiedler, Bernold; Rocha, Carlos; Wolfrum, Matthias

doi:10.1016/j.jde.2003.10.027

Cited by 29 publications

(58 citation statements)

References 35 publications

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“…Indeed, we recall that a frozen wave is never hyperbolic as a non-homogeneous equilibrium point of (1.2) (and not as a wave, as it was considered in the above lemmas). As already noticed in [13], if we replace f (u, u x ) by f (u, u x ) + εu x , we can "unfreeze" every frozen wave. To be sure that we are not "freezing" some rotating wave, we remark that there is at most a countable number of hyperbolic waves.…”

Section: Generic Hyperbolicity For Non-linearities Independent Of Xmentioning

confidence: 83%

“…Indeed, in [10], Fiedler and Mallet-Paret have proved that Equation (1.1) satisfies a Poincaré-Bendixson type property. For non-linearities independent of x, automatic transversality properties have been proved in [13], by showing that, for (1.2), the Morse-Smale property is equivalent to the hyperbolicity of all closed orbits. The results of the present paper thus imply the genericity of Morse-Smale property for Equation (1.2).…”

Section: Introductionmentioning

confidence: 99%

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Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle

Joly

Raugel

2010

Trans. Amer. Math. Soc.

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Abstract. In this paper, we show that, for scalar reaction-diffusion equations on the circle S 1 , the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non-linearity. In other words, we prove that in an appropriate functional space of non-linear terms in the equation, the set of functions, for which all equilibria and periodic orbits are hyperbolic, is a countable intersection of open dense sets. The main tools in the proof are the property of the lap number and the Sard-Smale Theorem.

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Section: Generic Hyperbolicity For Non-linearities Independent Of Xmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle

Joly

Raugel

2010

Trans. Amer. Math. Soc.

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“…This completes our sketch of a proof of Theorem 2.2.7. For complete details see [Fi&al02b]. We conclude this section with a few remarks concerning related results.…”

Section: Sturm Attractors On the Circlementioning

confidence: 90%

Spatio-Temporal Dynamics of Reaction-Diffusion Patterns

Fiedler

Scheel

2003

Trends in Nonlinear Analysis

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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...

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“…Spatial inhomogeneity is perfectly acceptable, even in applications to the existence of time-periodic orbits. This is a stark departure from analytic methods [5,16], which are efficacious in the setting of rigid rotating waves. The topological approach finds non-rigidly-rotating waves.…”

Section: Proofs: Forcing Periodic Solutionsmentioning

confidence: 99%

“…(1.1), e.g. [15,17,18,19,16], in that we focus on forcing on individual stationary or periodic orbits. Our methods are not used with the goal of characterizing the global attractor.…”

Section: Proofs: Forcing Periodic Solutionsmentioning

confidence: 99%

Scalar parabolic PDEs and braids

Vandervorst

2008

Trans. Amer. Math. Soc.

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Abstract. The comparison principle for scalar second order parabolic PDEs on functions u(t, x) admits a topological interpretation: pairs of solutions, u 1 (t, ·) and u 2 (t, ·), evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions {u α (t, ·)} n α=1 . By lifting the graphs to Legendrian braids, we give a global version of the comparison principle: the curves u α (t, ·) evolve so as to (weakly) decrease the algebraic length of the braid.We define a Morse-type theory on Legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic PDEs. This is done via discretization to a finite-dimensional system and a suitable Conley index for discrete braids.The result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic PDEs. We give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time-periodic solutions.

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Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

Cited by 29 publications

References 35 publications

Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle

Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle

Spatio-Temporal Dynamics of Reaction-Diffusion Patterns

Scalar parabolic PDEs and braids

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