Heteroclinic Orbits of Semilinear Parabolic Equations
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This article is published in Journal of Differential Equations.The article was published on 1996-02-10 and is currently open access. It has received 127 citations till now. The article focuses on the topics: Heteroclinic orbit & Parabolic partial differential equation.read more
Citations
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Topological methods in hydrodynamics
Vladimir I. Arnold,Boris Khesin +1 more
TL;DR: A group theoretical approach to hydrodynamics is proposed in this article, where the authors consider the hydrodynamic geometry of diffeomorphism groups and the principle of least action implies that the motion of a fluid is described by geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy.
Book ChapterDOI
Chapter 17 - Global Attractors in Partial Differential Equations
TL;DR: In this paper, the robustness and attractivity properties of global attractors in PDEs have been discussed, and weaker concepts of comparison, such as estimates of the Hausdorff distance between the global attractor distances, are considered.
Book ChapterDOI
Spatio-Temporal Dynamics of Reaction-Diffusion Patterns
Bernold Fiedler,Arnd Scheel +1 more
TL;DR: In this article, a survey of parabolic partial differential equations from a dynamical systems point of view is presented, where the success of dynamical concepts such as gradient flows, invariant manifolds, ergodicity, shift dynamics, etc.
Journal ArticleDOI
Recent developments in dynamical systems: three perspectives
TL;DR: This paper aims to an present account of some problems considered in the past years in Dynamical Systems, new research directions and also provide some open problems.
Book ChapterDOI
Chapter 16 - Parabolic Equations: Asymptotic Behavior and Dynamics on Invariant Manifolds
TL;DR: In this article, the authors discuss the asymptotic behavior and dynamics of parabolic equations on invariant manifolds and show that the maximum principle has long been used for various purposes in the study of Parabolic equations.
References
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Geometric Theory of Semilinear Parabolic Equations
TL;DR: The neighborhood of an invariant manifold near an equilibrium point is a neighborhood of nonlinear parabolic equations in physical, biological and engineering problems as mentioned in this paper, where the neighborhood of a periodic solution is defined by the invariance of the manifold.
Book
Shock Waves and Reaction-Diffusion Equations
TL;DR: In this paper, the basics of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley, are presented in a way accessible to a wider audience than just mathematicians.
Book
Asymptotic Behavior of Dissipative Systems
TL;DR: In this article, the authors consider a continuous dynamical system with a global attractor and describe the properties of the flow on the attractor asymptotically smooth and Morse-Smale maps.
MonographDOI
Isolated Invariant Sets and the Morse Index
TL;DR: On stable properties of the solution set of an ordinary differential equation, see as mentioned in this paper and the Morse index continuuation bibliography for a complete survey of the literature on flow stability and flow properties.
Book
Attractors for Semi-groups and Evolution Equations
TL;DR: The number of determining modes and the fractal dimension of bounded invariant sets for the Navier-Stokes equations were estimated in this paper, where the authors also considered the evolution equations of hyperbolic type.