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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Morse decomposition of global attractors with infinite components

Pages: 2845 - 2861, Volume 35, Issue 7, July 2015      doi:10.3934/dcds.2015.35.2845

 
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Tomás Caraballo - Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain (email)
Juan C. Jara - Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain (email)
José A. Langa - Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain (email)
José Valero - Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche, Spain (email)

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.

Keywords:  Morse decomposition, infinite components, gradient dynamics, Lyapunov function, gradient-like semigroup.
Mathematics Subject Classification:  37B25, 37L99, 35B40, 35B41.

Received: December 2013;      Revised: December 2014;      Available Online: January 2015.

 References