Chaos: Concepts, Control and Constructive Use (Understanding Complex Systems)

The study of physics has changed in character, mainly due to the passage from the analyses of linear systems to the analyses of nonlinear systems. Such a change began, it goes without saying, a long time ago but the qualitative change took place and boldly evolved after the understanding of the nature of chaos in nonlinear s- tems. The importance of these systems is due to the fact that the major part of physical reality is nonlinear. Linearity appears as a result of the simpli?cation of real systems, and often, is hardly achievable during the experimental studies. In this book, we focus our attention on some general phenomena, naturally linked with nonlinearity where chaos plays a constructive part. The ?rst chapter discusses the concept of chaos. It attempts to describe the me- ing of chaos according to the current understanding of it in physics and mat- matics. The content of this chapter is essential to understand the nature of chaos and its appearance in deterministic physical systems. Using the Turing machine, we formulate the concept of complexity according to Kolmogorov. Further, we state the algorithmic theory of Kolmogorov–Martin-Lof ¨ randomness, which gives a deep understanding of the nature of deterministic chaos. Readers will not need any advanced knowledge to understand it and all the necessary facts and de?nitions will be explained.

The study of chaotic behaviour in nonlinear, dynamical systems is now a well established research domain with ramifications into all fields of sciences, spanning a vast range of applications, from celestial mechanics, via climate change, to the functioning of brownian motors in cells.

A more recent discovery is that chaos can be controlled and, under appropriate conditions, can actually be constructive in the sense of becoming a control parameter itself for the system under investigation, stochastic resonance being a prime example.

The present work is putting emphasis on the latter aspects, and after recalling the paradigm changes introduced by the concept of chaos, leads the reader skillfully through the basics of chaos control by detailing relevant algorithms for both Hamiltonian and dissipative systems amongst others. 

The main part of the book is then devoted to the issue of synchronization in chaotic systems, an introduction to stochastic resonance and a survey of ratchet models.

This short and concise primer is particularly suitable for postgraduate students and non-specialist scientists from related areas, wishing to enter the field quickly and efficiently.