Advances in Difference Equations
Volume 2006 (2006), Article ID 23939, 15 pages
doi:10.1155/ADE/2006/23939
Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems
1Department of Mathematics, West University of Timişoara, Bd. V. Parvan 4, Timişoara 300223, Romania
2LAGA, UMR 7539, Institut Galilée, Université Paris 13, 99 Avenue J.B. Clément, Villetaneuse 93430, France
Received 15 October 2004; Accepted 18 October 2004
Copyright © 2006 St. Balint et al. 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.
Abstract
A method for determination and two methods for approximation of
the domain of attraction Da(0) of the asymptotically stable
zero steady state of an autonomous, ℝ-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 Da(0). The first method
of approximation uses a sequence of Lyapunov functions Vp, which converge to the Lyapunov function V on Da(0). Each Vp defines an estimate Np of Da(0). For any x∈Da(0), there exists an estimate Npx which contains x. The second method of approximation uses a ball B(R)⊂Da(0) which generates the sequence of estimates Mp=f−p(B(R)). For any x∈Da(0), there exists an estimate Mpx which contains x. The cases ‖∂0f‖<1 and ρ(∂0f)<1≤‖∂0f‖ are treated separately because significant
differences occur.