For over 100 years, ever since the groundbreaking works by H. Poincaré and A. Lyapunov, dynamical systems have remained a critical tool for investigating the behavior of
complex systems. The techniques introduced by Poincaré and Lyapunov were followed by inspiring discoveries by A. Kolmogorov, V. Arnold, J. Moser, S. Smale, and E. Lorenz, each motivated by challenges from a segment of physics. This book is an excellent and balanced introduction to dynamical systems, where the basic tenets of differential equations, such as questions related to existence, uniqueness, and stability, are developed in parallel with the approach one needs to address complex behavior in systems that display chaotic behavior.
The book consists of nine chapters and five appendices. The first six chapters are dedicated to the classical concepts of ordinary differential equations, ending with a thorough description of the phase plane analysis of two-dimensional flows. In the remaining three chapters, the basic theorems and algorithms of chaotic dynamics, bifurcation theory, and Hamiltonian dynamics are presented. While special attention is paid in the first six chapters to providing streamlined proofs for the great majority of the theorems, the approach in the last three chapters is to make sure the reader appreciates the motivation behind the concepts, and the resulting powerful computational tools. The reader is often encouraged to consult references to complete the investigation of concepts, such as Lyapunov exponents, Melnikov’s method, or Kolmogorov, Arnold, Moser (KAM) theory.
The appendices provide a snapshot of how software packages such as Mathematica, Maple, or MATLAB can be incorporated in one’s attempt to understand chaotic behavior. This segment of the text is sparse, because, understandably,
so many other texts exist whose (often sole) strength is in introducing the use of these powerful software packages. This is not to give the wrong impression that the text is not modern; there are in fact plenty of insightful figures that clearly enhance the reader’s experience. It is, however, clear that the author’s first instinct is to give readers an intuitively complete introduction to the foundational mathematics that characterize the current challenges in dynamical systems.
This book is written by a renowned teacher and practitioner of dynamical systems. It is based on the author’s decade-long experience in introducing this material to upper-level undergraduate and first-year graduate students. Although there are several other good treatments of this subject, it is safe to predict that this book will soon replace many others as the book of choice for upper-division students, especially those with relatively strong analytical backgrounds in mathematics.