On the structure ofthe global attractor for non-autonomous dynamical systems with weak convergence

Tomás Caraballo; David Cheban

doi:10.3934/cpaa.2012.11.809

Article Contents

2012, Volume 11, Issue 2: 809-828. Doi: 10.3934/cpaa.2012.11.809

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On the structure of the global attractor for non-autonomous dynamical systems with weak convergence

Tomás Caraballo^1, and
David Cheban^2,

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla
2.
State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău

Received: December 31, 2010

Revised: December 31, 2010

Published: February 2012

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Abstract

The aim of this paper is to describe the structure of global attractors for non-autonomous dynamical systems with recurrent coefficients (with both continuous and discrete time). We consider a special class of this type of systems (the so--called weak convergent systems). It is shown that, for weak convergent systems, the answer to Seifert's question (Does an almost periodic dissipative equation possess an almost periodic solution?) is affirmative, although, in general, even for scalar equations, the response is negative. We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our paper to the study of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations.

Keywords:

Non-autonomous dynamical systems,
skew-product systems,
cocycles,
global attractor,
dissipative systems,
convergent systems,
quasi-periodic,
almost periodic,
almost automorphic,
recurrent solutions,
asymptotically almost periodic solutions.

Mathematics Subject Classification: Primary: 34C11, 34C27, 34D05, 34D23, 34D45, 34K14,37B20, 37B55, 37C55, 7C60, 37C65, 37C70, 37C75.

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