Numerical computation of dichotomy rates and projectors in discrete time

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July 2009, 12(1): 109-131. doi: 10.3934/dcdsb.2009.12.109

Numerical computation of dichotomy rates and projectors in discrete time

1.	Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld

Received September 2008 Revised December 2008 Published May 2009

Abstract
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We introduce a characterization of exponential dichotomies for linear difference equations that can be tested numerically and enables the approximation of dichotomy rates and projectors with high accuracy. The test is based on computing the bounded solutions of a specific inhomogeneous difference equation. For this task a boundary value and a least squares approach is applied. The results are illustrated using Hénon's map. We compute approximations of dichotomy rates and projectors of the variational equation, along a homoclinic orbit and an orbit on the attractor as well as for an almost periodic example. For the boundary value and the least squares approach, we analyze in detail errors that occur, when restricting the infinite dimensional problem to a finite interval.

Keywords: Boundary value problem, Dichotomy projectors, Exponential dichotomy, Least squares solution..

Mathematics Subject Classification: Primary: 34D09, 37C75; Secondary: 34B0.

Citation: Thorsten Hüls. Numerical computation of dichotomy rates and projectors in discrete time. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 109-131. doi: 10.3934/dcdsb.2009.12.109

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