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


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.


Authors’ Affiliations

Department of Mathematics, West University of Timişoara
LAGA, UMR 7539, Institut Galilée


  1. Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985:xiii+561.View ArticleMATHGoogle Scholar
  2. Kaslik E, Balint AM, Birauas S, Balint St: Approximation of the domain of attraction of an asymptotically stable fixed point of a first order analytical system of difference equations. Nonlinear Studies 2003,10(2):103–112.MathSciNetMATHGoogle Scholar
  3. Kaslik E, Balint AM, Grigis A, Balint St: An extension of the characterization of the domain of attraction of an asymptotically stable fixed point in the case of a nonlinear discrete dynamical system. In Proceedings of 5th ICNPAA. Edited by: Sivasundaram S. European Conference Publications, Cambridge, UK; 2004.Google Scholar
  4. Kelley WG, Peterson AC: Difference Equations. 2nd edition. Harcourt/Academic Press, California; 2001:x+403.MATHGoogle Scholar
  5. Koçak H: Differential and Difference Equations through Computer Experiments. 2nd edition. Springer, New York; 1989:xviii+224.View ArticleMATHGoogle Scholar
  6. Ladas G, Qian C, Vlahos PN, Yan J: Stability of solutions of linear nonautonomous difference equations. Applicable Analysis. An International Journal 1991,41(1–4):183–191. 10.1080/00036819108840023MathSciNetView ArticleMATHGoogle Scholar
  7. Lakshmikantham V, Trigiante D: Theory of Difference Equations. Numerical Methods and Applications, Mathematics in Science and Engineering. Volume 181. Academic Press, Massachusetts; 1988:x+242.Google Scholar
  8. LaSalle JP: The Stability and Control of Discrete Processes, Applied Mathematical Sciences. Volume 62. Springer, New York; 1986:vi+150.View ArticleGoogle Scholar
  9. LaSalle JP: Stability theory for difference equations. In Studies in Ordinary Differntial Equations, MAA Studies in Mathematics. Volume 14. Edited by: Hale J. Taylor and Francis Science Publishers, London; 1997:1–31.Google Scholar


© Hindawi Publishing Corporation. 2006

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