Regular and Chaotic Oscillations

In this book the modern theory of both regular and chaotic nonlinear oscillations is set out, primarily, as applied to mechanical problems. The material is presented in a nontraditional manner with emphasis on the new results of the theory obtained partially by the author, who is one of the leading experts in the area. Among the up-to-date topics are synchronization and chaotization of self-oscillatory systems and the influence of weak random vibrations on the modification of characteristics and behavior of nonlinear systems. One of the purposes of the book is to enable readers to gain a thorough understanding of this theory and to show that it can be very useful in engineering investigations. The primary audience for this book is researchers working with different oscillatory processes and students interested in a thorough study of the general laws and applications of the theory of nonlinear 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.