Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 539087, 23 pages
doi:10.1155/2010/539087
Research Article

Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

Josef Diblík,1,2 Denys Ya. Khusainov,3 Irina V. Grytsay,3 and Zdenĕk Šmarda1

1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic
2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 60200 Brno, Czech Republic
3Department of Complex System Modeling, Faculty of Cybernetics, Taras, Shevchenko National University of Kyiv, Vladimirskaya Str., 64, 01033 Kyiv, Ukraine

Received 28 January 2010; Accepted 11 May 2010

Academic Editor: Elena Braverman

Copyright © 2010 Josef Diblík 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

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalue λ=1 of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.