A simple method for studying asymptotic stability of discrete dynamical systems and its applications

An International Journal of Optimization and Control: Theories & Applications

article

Manh Tuan Hoang¹  Thi Kim Quy Ngo²  Ha Hai Truong³ 

[1] Department of Mathematics, FPT University;Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology;Department of Basic Sciences, Thai Nguyen University of Information and Communication Technology

关键词: Discrete dynamical systems;    Lyapunov's indirect method;    Asymptotic stability;    Non-hyperbolic equilibrium point;    Nonstandard finite difference methods;   

DOI  :  10.11121/ijocta.2023.1243

学科分类：地球科学(综合)

来源: Balikesir University

【 摘 要 】

In this work, we introduce a simple method for investigating the asymptotic stability of discrete dynamical systems, which can be considered as an extension of the classical Lyapunov's indirect method. This method is constructed based on the classical Lyapunov's indirect method and the idea proposed by Ghaffari and Lasemi in a recent work. The new method can be applicable even when equilibia of dynamical systems are non-hyperbolic. Hence, in many cases, the classical Lyapunov's indirect method fails but the new one can be used simply. In addition, by combining the new stability method with the Mickens' methodology, we formulate some nonstandard finite difference (NSFD) methods which are able to preserve the asymptotic stability of some classes of differential equation models even when they have non-hyperbolic equilibrium points. As an important consequence, some well-known results on stability-preserving NSFD schemes for autonomous dynamical systems are improved and extended. Finally, a set of numerical examples are performed to illustrate and support the theoretical findings.

【 授权许可】

CC BY