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March  2007, 18(2&3): 499-515. doi: 10.3934/dcds.2007.18.499

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, Italy

Received  February 2006 Revised  July 2006 Published  March 2007

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, 39C5.
Citation: David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
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