Uniform nonautonomous attractors under discretization

P.E. Kloeden; Victor S. Kozyakin

doi:10.3934/dcds.2004.10.423

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2004, Volume 10, Issue 1&2: 423-433. Doi: 10.3934/dcds.2004.10.423

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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.
2.
Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane, 19, 101447 Moscow

Received: October 2001

Revised: February 2003

Published: January 2004

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Abstract

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.

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Mathematics Subject Classification: 34D10, 34D45, 37C60.

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