Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations

Henry, Daniel

doi:10.1016/0022-0396(85)90153-6

Cited by 201 publications

(159 citation statements)

References 6 publications

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“…Actually, for every λ ∈ (n 2 , (n + 1) 2 ) where n is any nonnegative integer, it was proved by Chafee and Infante in [17] that problem (4.1)-(4.3) has exactly 2n + 1 equilibrium solutions. It was further proved by Henry in [18] that all these equilibrium solutions are hyperbolic.…”

Section: Applicationsmentioning

confidence: 94%

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

Wang

2011

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodicsolutions. An application of the result to the Chafee-Infante equation is discussed.

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Section: Applicationsmentioning

confidence: 94%

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

Wang

2011

Nonlinear Analysis: Theory, Methods & Applications

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“…The significance of non-pitchforkable Sturm attractors was that their heteroclinic orbits are not accessible to analysis via successive pitchfork bifurcations. Therefore the original approach by Conley, Smoller and Henry [CoSm83,He85] to the heteroclinic orbit problem failed for these Sturm attractors. We caution the combinatorially and dynamically well-versed reader that edge deletion and edge contraction on bi-polar 1-skeleton C 1 f as discussed in [FMR95], section 6 does not, in general, correspond to pitchfork bifurcations in the associated Sturm attractor A f .…”

Section: Discussionmentioning

confidence: 99%

“…It was a celebrated result of Angenent and Henry, independently, that this Morse-Smale transversality is, not an additional requirement but, a consequence of hyperbolicity of equilibria; see [He85,An86]. Surprisingly this fact is based on a generalization of the Sturm nodal property, first observed by [St1836] and very successfully revived by [Ma82].…”

Section: Introductionmentioning

confidence: 99%

Connectivity and Design of Planar Global Attractors of Sturm Type. III: Small and Platonic Examples

Fiedler

Rocha

2009

J Dyn Diff Equat

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Based on a Morse-Smale structure we study planar global attractors A f of the scalar reaction-advection-diffusion equation u t = u xx + f (x, u, u x ) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f , and hyperbolicity of equilibria. We call A f Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type.The planar Sturm attractor consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph C f which we call the connection graph. Its 1-skeleton C 1 f consists of the unstable manifolds (separatrices) of the index-1 Morse saddles.We present two results which completely characterize the connection graphs C f and their 1-skeletons C In the present paper we show the equivalence of the two characterizations. Moreover we show that connection graphs of Sturm attractors indeed satisfy the required properties. In [FiRo07a] we show, conversely, how to design a planar Sturm attractor with prescribed plane connection graph or 1-skeleton of the required properties. In [FiRo07b] we describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons.

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“…We note that these authors do not use energy information, but instead rely on consequences of the maximum principle, which provide a very precise description of stable and unstable manifolds of equilibria. In particular, it is possible to prove [5,32] that the flows generated by such dynamical systems are Morse-Smale. This additional information allows one to rule out certain topologies of attractors.…”

Section: Remarks and Conclusionmentioning

confidence: 99%

The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor

Grinfeld

Novick-Cohen

1999

Trans. Amer. Math. Soc.

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Abstract. In this paper we establish a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn-Hilliard equation by explicit energy calculations. Strong non-degeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the flow on the global attractor is shown to be semi-conjugate to the flow on the global attractor of the Chaffee-Infante equation, and in the metastable case close to the nonlocal reaction-diffusion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle.

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Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations

Cited by 201 publications

References 6 publications

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

Connectivity and Design of Planar Global Attractors of Sturm Type. III: Small and Platonic Examples

The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor

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