Because of their inherent structure, complex systems exhibit considerably richer behavior than their underlying individual subsystems. This book is about new methods of mathematical analysis for time series generated by complex systems. Typical attributes of multiscale time series are: nonlinearity, sensitive dependence on small perturbations, long-time dependence on past behavior, and nonstationarity.

Gao et al. concentrate on the properties of two theories that have had substantial success: the chaos theory and the random fractal theory. They make a special effort to introduce these topics from the ground up and in the context of novel and fresh examples. The text is further strengthened by making a large set of real data available to the reader on a Web site that is kept current by the authors.

The book consists of 15 chapters and three appendices. The first seven chapters present the fundamental tenets of fractal and chaos theories, Fourier and wavelet representations, as well as a thorough introduction to probability theory. The substantive part of the book begins with chapter 7, where the synthesis of Brownian motion is described. The remaining chapters treat Levy motion, long-memory processes, multifractals and multiplicative processes, and power-law behavior of time series. In these chapters, especially the final one where the computation of scale-dependent Lyapunov exponents takes center stage, the authors pose important questions about chaos, noisy chaos, noise-induced chaos, and how one can distinguish among them (if not theoretically, at least practically). A large number of diverse examples invite the reader to continue exploring the book’s contents.

Bibliographical notes, appendices, and an excellent list of 498 references point the interested reader to a wide variety of further investigations.