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

Approximation of attractors of nonautonomous dynamical systems

Pages: 215 - 238, Volume 5, Issue 2, May 2005      doi:10.3934/dcdsb.2005.5.215

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Bernd Aulbach - Department of Mathematics, University of Augsburg, D-86135 Augsburg, Germany (email)
Martin Rasmussen - Department of Mathematics, University of Augsburg, D-86135 Augsburg, Germany (email)
Stefan Siegmund - Department of Mathematics, University of Frankfurt, D-60325 Frankfurt, Germany (email)

Abstract: This paper is devoted to the numerical approximation of attractors. For general nonautonomous dynamical systems we first introduce a new type of attractor which includes some classes of noncompact attractors such as unbounded unstable manifolds. We then adapt two cell mapping algorithms to the nonautonomous setting and use the computer program GAIO for the analysis of an explicit example, a two-dimensional system of nonautonomous difference equations. Finally we present numerical data which indicate a bifurcation of nonautonomous attractors in the Duffing-van der Pol oscillator.

Keywords:  Numerical approximation, Invariant manifold, Pullback attractor, Forward attractor, Nonautonomous diference equation, Nonautonomous dynamical system, Continuation algorithm, Subdivision algorithm, Nonautonomous bifurcation.
Mathematics Subject Classification:  34C30, 34D45, 37B55, 37G35, 39A10, 65L20.

Received: December 2003;      Revised: July 2004;      Available Online: February 2005.