Non-autonomous dynamical systems

Alexandre N. Carvalho; José A. Langa; James C. Robinson

doi:10.3934/dcdsb.2015.20.703

Article Contents

2015, Volume 20, Issue 3: 703-747. Doi: 10.3934/dcdsb.2015.20.703

This issue Previous Article Preface Next Article Pullback attractors for generalized evolutionary systems

Non-autonomous dynamical systems

1.
Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
3.
Mathematical Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL

Received: June 2014

Revised: September 2014

Published: May 2015

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Abstract

This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors. All of them are closely related to the study of the pullback asymptotic limits of the dynamical system, from which naturally emerges the concept of a pullback attractor. In the first part of this paper we want to clarify these different dynamical scenarios and the relations between their corresponding attractors.
If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.

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Mathematics Subject Classification: 37B55, 37L05, 37C70, 34D30, 37B35, 37D15.

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