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Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 4375069, 10 pages
http://dx.doi.org/10.1155/2016/4375069
Research Article

Approaching the Discrete Dynamical Systems by means of Skew-Evolution Semiflows

Department of Mathematics and Computer Science, Aurel Vlaicu University of Arad, 2 Elena Drăgoi Street, 310330 Arad, Romania

Received 10 April 2016; Accepted 9 June 2016

Academic Editor: Allan C. Peterson

Copyright © 2016 Codruţa Stoica. 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

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.