Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Uniform nonautonomous attractors under discretization

Pages: 423 - 433, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.423

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P.E. Kloeden - FB Mathematik, Johann Wolfgang Goethe Universit├Ąt, Postfach 11 19 32, D-60054 Frankfurt a.M., Germany (email)
Victor S. Kozyakin - Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane, 19, 101447 Moscow, Russian Federation (email)

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.

Keywords:  Cocycle dynamical systems, attractors, perturbations, discretization.
Mathematics Subject Classification:  34D10, 34D45, 37C60.

Received: October 2001;      Revised: February 2003;      Available Online: October 2003.