Topological Methods in Nonlinear Analysis

Continuity of Lyapunov functions and of energy level for a generalized gradient semigroup

Eder R. Aragão-Costa, Tomás Caraballo, Alexandre N. Carvalho, and José A. Langa

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Abstract

The global attractor of a gradient-like semigroup has a Morse decomposition. Associated to this Morse decomposition there is a Lyapunov function (differentiable along solutions) - defined on the whole phase space - which proves relevant information on the structure of the attractor. In this paper we prove the continuity of these Lyapunov functions under perturbation. On the other hand, the attractor of a gradient-like semigroup also has an energy level decomposition which is again a Morse decomposition but with a total order between any two components. We claim that, from a dynamical point of view, this is the optimal decomposition of a global attractor; that is, if we start from the finest Morse decomposition, the energy level decomposition is the coarsest Morse decomposition that still produces a Lyapunov function which gives the same information about the structure of the attractor. We also establish sufficient conditions which ensure the stability of this kind of decomposition under perturbation. In particular, if connections between different isolated invariant sets inside the attractor remain under perturbation, we show the continuity of the energy level Morse decomposition. The class of Morse-Smale systems illustrates our results.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 39, Number 1 (2012), 57-82.

Dates
First available in Project Euclid: 20 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461184854

Mathematical Reviews number (MathSciNet)
MR2934334

Zentralblatt MATH identifier
1271.37019

Citation

Aragão-Costa, Eder R.; Caraballo, Tomás; Carvalho, Alexandre N.; Langa, José A. Continuity of Lyapunov functions and of energy level for a generalized gradient semigroup. Topol. Methods Nonlinear Anal. 39 (2012), no. 1, 57--82. https://projecteuclid.org/euclid.tmna/1461184854


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