The discovery of many nonlinear phenomena and their study by systematic methods are a major breakthrough in science and mathematics of the 20th Century, leading to the research and development of nonlinear science, which is at the forefront of science and technology of the 21st Century. Chaos is an extreme form of nonlinear dynamical phenomena. But what exactly is chaos? This is the main focus of this book. Mathematical definitions of chaos can be given in many different ways. Though we will give the first of such definitions at the end of Chap.  2 (in Definition.  2.7 ), during much of the first few chapters the term “chaos” (or its adjective “chaotic”) should be interpreted in a rather liberal and intuitive sense that it stands for some irregular behaviors or phenomena. This vagueness should automatically take care of itself once more rigorous definitions are given.