Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems

David Cheban; Cristiana Mammana

doi:10.3934/dcds.2007.18.499

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2007, Volume 18, Issue 2&3: 499-515. Doi: 10.3934/dcds.2007.18.499

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Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems

David Cheban^1, and
Cristiana Mammana^2,

1.
State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău
2.
Institute of Economics and Finances, University of Macerata, str. Crescimbeni 14, I–62100 Macerata

Received: February 2006

Revised: July 2006

Published: June 2007

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Abstract

The paper is dedicated to the study of the problem of continuous dependence of compact global attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems (IIFS). We prove that if a family of non-autonomous dynamical systems ‹$(X,\mathbb T_1,\pi_{\lambda}),(Y,\mathbb T_{2},\sigma),h $›depending on parameter $\lambda\in\Lambda$ is uniformly contracting (in the generalized sense), then each system of this family admits a compact global attractor $J^{\lambda}$ and the mapping $\lambda \to J^{\lambda}$ is continuous with respect to the Hausdorff metric. As an application we give a generalization of well known Theorem of Bransley concerning the continuous dependence of fractals on parameters.

Keywords:

Global attractor; non-autonomous dynamical system; infinite iterated functions systems.

Mathematics Subject Classification: 37B25, 37B55, 39A11, 39C10, 39C55.

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