Abstract
In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.
Highlights
The asymptotic behaviour of a system of differential equations modeling real phenomena from different areas of Science is usually described by the analysis of their global attractors, a compact invariant set for the associated semigroups attracting bounded sets forwards in time
Vis called a Lyapunov function related to M. It has been proved in [1] that given a disjoint family of isolated invariant sets on the global attractor M = {M1, · · ·, Mn} for a semigroup T (t), the dynamical property of being generalized dynamically gradient, the existence of an associated ordered family of local attractor-repellers, and the existence of a Lyapunov functional related to M, are equivalent properties
5 and 6 we prove the main result of this paper, the equivalence between a generalized dynamically gradient semigroup referred to M∞ with a suitable separation assumption, the existence of an ordered Lyapunov function associated to M∞, and the existence of a Morse decomposition on the global attractor. This is done in several steps: first, we prove that the property of the semigroup of being generalized dynamically gradient together with a separation assumption implies that a Morse decomposition can be constructed; we prove that from a Morse decomposition related to M∞ an ordered Lyapunov function can be defined; we check that the existence of an ordered Lyapunov function implies that the semigroup is generalized dynamically gradient semigroup referred to M∞ and that the separation assumption holds
Summary
The asymptotic behaviour of a system of (ordinary or partial) differential equations modeling real phenomena from different areas of Science is usually described by the analysis of their global attractors, a compact invariant set for the associated semigroups attracting (uniformly) bounded sets forwards in time.
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