Lyapunov functions and  attractors under variable  time-step discretization

Peter E. Kloeden; Björn Schmalfuss

doi:10.3934/dcds.1996.2.163

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1996, Volume 2, Issue 2: 163-172. Doi: 10.3934/dcds.1996.2.163

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Lyapunov functions and attractors under variable time-step discretization

Peter E. Kloeden^1, and
Björn Schmalfuss^2,

1.
School of Computing and Mathematics, Deakin University, Geelong, Victoria 3217
2.
Institut für Dynamische Systeme, Universität Bremen, D-28334 Bremen

Received: January 31, 1995

Revised: September 30, 1995

Published: March 1996

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Abstract

A one-step numerical scheme with variable time--steps is applied to an autonomous differential equation with a uniformly asymptotically stable set, which is compact but otherwise of arbitrary geometric shape. A Lyapunov function characterizing this set is used to show that the resulting nonautonomous difference equation generated by the numerical scheme has an absorbing set. The existence of a cocycle attractor consisting of a family of equivariant sets for the associated discrete time cocycle is then established and shown to be close in the Hausdorff separation to the original stable set for sufficiently small maximal time-steps.

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Mathematics Subject Classification: 34C35, 65L05.

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