Open Access

Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems

Advances in Difference Equations20062006:023939

DOI: 10.1155/ADE/2006/23939

Received: 15 October 2004

Accepted: 18 October 2004

Published: 12 February 2006

Abstract

A method for determination and two methods for approximation of the domain of attraction D a (0) of the asymptotically stable zero steady state of an autonomous, -analytical, discrete dynamical system are presented. The method of determination is based on the construction of a Lyapunov function V, whose domain of analyticity is D a (0). The first method of approximation uses a sequence of Lyapunov functions V p , which converge to the Lyapunov function V on D a (0). Each V p defines an estimate N p of D a (0). For any x D a (0), there exists an estimate which contains x. The second method of approximation uses a ball B(R) D a (0) which generates the sequence of estimates M p = f-p(B(R)). For any x D a (0), there exists an estimate which contains x. The cases ||∂0f||<1 and ρ(∂0f) < 1 ≤||∂0f|| are treated separately because significant differences occur.

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Authors’ Affiliations

(1)
Department of Mathematics, West University of Timişoara
(2)
LAGA, UMR 7539, Institut Galilée

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Copyright

© Hindawi Publishing Corporation. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.