Application of Stability Theory in Study of Local Dynamics of Nonlinear Systems

Abstract

Investigating local dynamics of equilibrium points of nonlinear systems plays an important role in studying the behavior of dynamical systems. There are many different definitions for stable and unstable solutions in the literature. The main goal to develop stability definitions is exploring the responses or output of a system to perturbation as time approaches infinity. Due to the wide range of application of local dynamical system theory in physics, biology, economics and social science, it still attracts many researchers to play with its definitions to find out the answers for their questions. In this paper, we start with a brief review over continuous time dynamical systems modeling and then we bring useful examples to the playground. We study the local dynamics of some interesting systems and we show the local stable behavior of the system around its critical points. Moreover, we look at local dynamical behavior of famous dynamical systems, Hénon-Heiles system, Duffing oscillator and Van der Pol equation and analyze them. Finally, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples.