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February  2004, 10(1&2): 423-433. doi: 10.3934/dcds.2004.10.423

Uniform nonautonomous attractors under discretization

P.E. Kloeden 1, and  Victor S. Kozyakin 2,

1. 

FB Mathematik, Johann Wolfgang Goethe Universität, Postfach 11 19 32, D-60054 Frankfurt a.M., Germany
2. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane, 19, 101447 Moscow, Russian Federation

Received  October 2001 Revised  February 2003 Published  October 2003

A nonautonomous or cocycle dynamical system that is driven by an autonomous dynamical system acting on a compact metric space is assumed to have a uniform pullback attractor. It is shown that discretization by a one-step numerical scheme gives rise to a discrete time cocycle dynamical system with a uniform pullback attractor, the component subsets of which converge upper semi continuously to their continuous time counterparts as the maximum time step decreases to zero. The proof involves a Lyapunov function characterizing the uniform pullback attractor of the original system.
Keywords: perturbations, discretization., Cocycle dynamical systems, attractors.
Mathematics Subject Classification: 34D10, 34D45, 37C6.
Citation: P.E. Kloeden, Victor S. Kozyakin. Uniform nonautonomous attractors under discretization. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 423-433. doi: 10.3934/dcds.2004.10.423
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