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

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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Correspondence to St Balint.

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Balint, S., Kaslik, E., Balint, A. et al. Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems. Adv Differ Equ 2006, 023939 (2006). https://doi.org/10.1155/ADE/2006/23939

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Keywords

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