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  • Research Article
  • 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

https://doi.org/10.1155/ADE/2006/23939

  • Received: 15 October 2004
  • Accepted: 18 October 2004
  • Published:

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.

Keywords

  • Differential Equation
  • Steady State
  • Dynamical System
  • Partial Differential Equation
  • Ordinary Differential Equation

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Authors’ Affiliations

(1)
Department of Mathematics, West University of Timişoara, Bd. V. Parvan 4, Timişoara, 300223, Romania
(2)
LAGA, UMR 7539, Institut Galilée, Université Paris 13, 99 Avenue J.B. Clément, Villetaneuse, 93430, France

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