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March  2007, 18(2&3): i-i. doi: 10.3934/dcds.2007.18.2i

Preface

Tomás Caraballo 1, , Amadeu Delshams 2, , Àngel Jorba 3, , Peter E. Kloeden 4, and  Rafael Obaya 5,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2. 

Departament de Matemàtica Aplicada I, ETSEIB-Universitat Politècnica de Catalunya, Diagonal 647, E-08028 Barcelona
3. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585 , 08007 Barcelona
4. 

Institut für Computerorientierte Mathematik, J.W. Goethe Universität, 60054 Frankfurt, Germany
5. 

Universidad de Valladolid, Departamento de Matemática Aplicada, ETSII, Paseo del Cauce s/n, 47011 Valladolid

Published  March 2007

The theory of dynamical systems has undergone some spectacular and fascinating developments in the past century, as the readers of this journal are well aware, with the focus predominately on autonomous systems. There are many ways in which one could classify the work that has been done, but one that stands clearly in the forefront is the distinction between dissipative systems with their attractors and conservative systems, in particular Hamiltonian systems.
    Another classification is between autonomous and nonautonomous systems. Of course, the latter subsumes the former as special case, but with the former having special structural features, i.e., the semigroup evolution property, which has allowed an extensive and seemingly complete theory to be developed. Although not as extensive, there have also been significant developments in the past half century on nonautonomous dynamical systems, in particular the skew-product formalism involving a cocycle evolution property which generalizes the semigroup property of autonomous systems. This has been enriched in recent years by advances on random dynamical systems, which are roughly said a measure theoretic version of a skew-product flow. In particular, new concepts of random and nonautonomous attractors have been introduced and investigated.

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Citation: Tomás Caraballo, Amadeu Delshams, Àngel Jorba, Peter E. Kloeden, Rafael Obaya. Preface. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : i-i. doi: 10.3934/dcds.2007.18.2i
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