Study on the Application of Stability Theory in Studying the Local Dynamics of Nonlinear Systems

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. We assumed a Hamiltonian function with two degrees
of freedom and it can be obtained by adding an integrable Hamiltonian system and a non-integrable
Hamiltonian system. We showed that for = 0 and also for 0 < 1, there exist quasi periodic
cycles which are known as KAM tori. However, under perturbation, these quasi periodic cycles will be
deformed and KAM tori will be dissolved gradually as we increase E.