Chapter 2

Introduction to Chaos Theory and Electric Drive Systems

As this book is a happy marriage between two disciplines – chaos theory and electric drive systems – the coverage of knowledge is so broad that it is desirable to briefly introduce basic chaos theory and the fundamentals of electric drive systems.

In this chapter, the necessary background knowledge of this book, namely the basic theory of chaos and some fundamentals of electric drive systems, are discussed.

2.1 Basic Chaos Theory

This section describes some of the basic principles of chaos, including the concept of dynamical systems, discrete maps, limit sets, attractors, stability, and manifolds. The criteria of chaos are then illustrated using the Lyapunov exponents, fractal dimensions, and entropy. Hence, bifurcations and routes to chaos will both be discussed. Finally, various methods for chaos analysis are introduced, which include waveforms, phase portraits, the Poincaré map, bifurcation diagrams, time-series reconstruction, and the calculations of Lyapunov exponents, embedded unstable periodic orbits, power spectra, fractal dimensions, and entropy.

2.1.1 Basic Principles

2.1.1.1 Dynamical Systems

Autonomous Dynamical Systems

An nth-order autonomous dynamical system is defined by , . The corresponding solution ...