Regular and Chaotic Oscillations

1. Introduction.- 1.1 The importance of oscillation theory for engineering mechanics.- 1.2 Classification of dynamical systems. Systems with conservation of phase volume and dissipative systems.- 1.3 Different types of mathematical models and their functions in studies of concrete systems.- 1.4 Phase space of autonomous dynamical systems and the number of degrees of freedom.- 1.5 The subject matter of the book.- 2. The main analytical methods of studies of nonlinear oscillations in near-conservative systems.- 2.1 The van der Pol method.- 2.2 The asymptotic Krylov-Bogolyubov method.- 2.3 The averaging method.- 2.4 The averaging method in systems incorporating fast and slow variables.- 2.5 The Whitham method.- I. Oscillations in Autonomous Dynamical Systems.- 3. General properties of autonomous dynamical systems.- 4. Examples of natural oscillations in systems with one degree of freedom.- 5. Natural oscillations in systems with many degrees of freedom. Normal oscillations.- 6. Self-oscillatory systems with one degree of freedom.- 7. Self-oscillatory systems with one and a half degrees of freedom.- 8. Examples of self-oscillatory systems with two or more degrees of freedom.- 9. Synchronization and chaotization of self-oscillatory systems by an external harmonic force.- 10. Interaction of two self-oscillatory systems. Synchronization and chaotization of self-oscillations.- 11. Interaction of three or more self-oscillatory systems.- II. Oscillations in Nonautonomous Systems.- 12. Oscillations of nonlinear systems excited by external periodic forces.- 13. Parametric excitation of oscillations.- 14. Changes in the dynamical behavior of nonlinear systems induced by high-frequency vibration or by noise.- A. Derivation of the approximate equation for the one-dimensional probability density.- References.

Mechanics-oriented monograph from one of the leading Russian physicists. Most of the established books in this field (Nayfeh, Minorski) are out of date and do not regard the tremendous developments of nonlinear dynamics in the past 20 years.

From the reviews of the first edition:



"The present book ... gives an up to date overview of recent developments, especially focusing on work done in the countries of the former Soviet Union. In this respect it is a gold mine for readers who have already some similarity with the theory of nonlinear oscillations, because 375 references are listed. ... For those readers who want to get an overview on modern developments of oscillation theory and their applications in the sciences, the book will be certainly an excellent choice." (H. Troger, Zentralblatt MATH, Vol. 980, 2002)