Chaos Suppression with Least Prior Knowledge: Continuous Time Feedback

Abstract

Firstly, we present the frequency spectrum as an alternative procedure for studying feedback effects onto chaotic systems. It is well known that continuous power spectrum is an important feature of chaotic systems. This fact can be used for distinguishing a chaotic system from time series. Although power spectrum is not definitive for identifying chaotic systems [1], power spectrum allows us to understand the effect of feedback onto chaotic systems in terms of the control parameters. In some sense, power spectrum can be seem as a dynamic bifurcation diagram [2], in fact, to study bifurcation of chaotic systems can be an important tool. To this end, there are two basic concepts: (i) Dynamics of a given nonlinear system can be approached by \(\dot{\chi}= f(\chi;\pi)\), where \(f: \mathbb{R}^{n}\rightarrow {\mathbb R}^{n}\) and \(\pi \in {\mathbb R}^{p}\) is a set of parameters (which can be a constant or time functions). Thus, qualitative changes of the system behavior can be induced for certain values of the parameters \(\pi \in {\mathbb R}^{p}\), i.e., it is possible that the nonlinear system displays chaos. (ii) such qualitative changes in dynamics of a nonlinear system can be observed due to parametric variations (i.e., bifurcation diagram). Hence, bifurcation diagrams are very important and can be also used as characterization procedure for chaotic systems.