Chaos and integrability in nonlinear dynamics : an introduction /

A very well written book on a classic subject, which starts from first principles and reaches into the very exciting developments of the last 15 to 20 years. Nonlinear dynamics is introduced as a problem in differential equations with a geometrical structure. By the third chapter the reader has been exposed to the main mathematical problem of classical mechanics and the small divisor problem, with a very good account given of the Kolmogorov-Arnold-Moser theorem. A brief but very informative account of the recent developments on the chaotic behavior of Hamiltonian systems is probably the first book treatment of this subject at a level accessible to advanced undergraduates or beginning graduate students. Tabor selects a few basic research papers from which he has drawn the material, provides the necessary but difficult-to-find background for understanding the papers, and gives simple illustrative examples that clarify the ideas. A brief treatment of dissipative dynamical systems provides a nice and well-placed methodological contrast to the Hamiltonian theory, as well as several important applications to fluid dynamics and the onset of turbulence. Three chapters dealing with partial differential equations are more difficult mathematically. The material covered is very broad, but again Tabor succeeds in conveying the role of chaos and instability in this more complex setting without excessive technicalities. Tabor brings together in a new mix topics that are usually separately and in widely different contexts, he makes contact with the current research literature without being excessively demanding, and he maintains throughout a well-balanced level of classical and modern mathematical techniques. An excellent book: highly recommended. -G. Papanicolaou, New York University