Bifurcations and chaos in simple dynamical systems

Abstract

Chaos is an active research subject in the fields of science in recent years. It is a complex and an erratic behaviour that is possible in very simple systems. In the present day, the chaotic behaviour can be observed in experiments. Many studies have been made in chaotic dynamics during the past three decades and many simple chaotic systems have been discovered. In this work, the behaviour of some simple dynamical systems is studied by constructing mathematical models. Investigations are made on the periodic orbits for continuous maps and idea of sensitive dependence on initial conditions, which is the hallmark of chaos, is obtained. A small attempt has been made to find out the reasons / unknown conditions for the production of chaos in a system. This is explained through simple dynamical systems. (Why chaos is produced in a forced damped simple pendulum?) Besides, another attempt has been made to identify, algebraically simplest chaotic flow. These are the significance of this study - “Bifurcations and chaos in simple dynamical systems”. Accordingly, an analysis is done on different dynamical systems. The exact solution is obtained by solving the differential equation by using Runge-Kutta method; as a result, it is clear from the analysis that, period multiplication occurring in a forced damped simple pendulum can leads to chaos. This result is proving that it is possible to find out the reasons / unknown conditions under which chaotic behaviour exhibits in various systems. The importance of result is, “why chaos is produced in various systems? - may be identified in future. Key Words: Bifurcations, chaos, dynamical systems.