Bifurcations and Onset of Chaos in Dissipative Systems

Abstract

In the earlier chapters, we have seen some basic features associated with nonlinear dynamical systems. Now we want to have a more detailed picture of the types of possible motions admitted by typical nonlinear systems, particularly by dissipative systems. Specifically, we wish to ask the question how does the dynamics change as a control parameter (like the strength of nonlinearity or the strength of the external forcing or its frequency) is smoothly varied. One finds that very interesting features arise in this process: Bifurcations, which are sudden qualitative changes in the nature of the motion, occur at critical control parameter values (as noted briefly in the earlier chapter). Often these bifurcations lead to chaotic behaviour of the dynamical system. How and why do these bifurcations occur? What are the mechanisms responsible for such bifurcations? How can they be classified? These are some of the important questions which arise naturally. One finds that they often occur in specific ways or routes as the control parameter is varied. Typically one type of motion loses stability at a critical value of the parameter as it varies smoothly giving rise to a new type of stable motion. This process can continue further to give rise to newer and newer motions. Thus a possible approach to understand the sudden qualitative changes in the dynamics is to look for the local stability properties of the solutions in the neighbourhood of the critical parameter values at which bifurcations occur.