Nonlinear Oscillations and Waves in Dynamical Systems

A rich variety of books devoted to dynamical chaos, solitons, self-organization has appeared in recent years. These problems were all considered independently of one another. Therefore many of readers of these books do not suspect that the problems discussed are divisions of a great generalizing science - the theory of oscillations and waves. This science is not some branch of physics or mechanics, it is a science in its own right. It is in some sense a meta-science. In this respect the theory of oscillations and waves is closest to mathematics. In this book we call the reader's attention to the present-day theory of non-linear oscillations and waves. Oscillatory and wave processes in the systems of diversified physical natures, both periodic and chaotic, are considered from a unified poin t of view . The relation between the theory of oscillations and waves, non-linear dynamics and synergetics is discussed. One of the purposes of this book is to convince reader of the necessity of a thorough study popular branches of of the theory of oscillat ions and waves, and to show that such science as non-linear dynamics, synergetics, soliton theory, and so on, are, in fact , constituent parts of this theory. The primary audiences for this book are researchers having to do with oscillatory and wave processes, and both students and post-graduate students interested in a deep study of the general laws and applications of the theory of oscillations and waves.

Inhalt
1 Dynamical systems. Phase space. Stochastic and chaotic systems. The number of degrees of freedom.- 2 Hamiltonian systems close to integrable. Appearance of stochastic motions in Hamiltonian systems.- 3 Attractors and repellers. Reconstruction of attractors from an experimental time series. Quantitative characteristics of attractors.- 4 Natural and forced oscillations and waves. Self-oscillations and auto-waves.- 5 Conservative systems.- 6 Non-conservative Hamiltonian systems and dissipative systems.- 7 Natural oscillations of non-linear oscillators.- 8 Natural oscillations in systems of coupled oscillators.- 9 Natural waves in bounded and unbounded continuous media. Solitons.- 10 Oscillations of a non-linear oscillator excited by an external force.- 11 Oscillations of coupled non-linear oscillators excited by an external periodic force.- 12 Parametric oscillations.- 13 Waves in semibounded media excited by perturbations applied to their boundaries.- 14 Forced oscillations and waves in active non-self-oscillatory systems. Turbulence. Burst instability. Excitation of waves with negative energy.- 15 Mechanisms of excitation and amplitude limitation of self-oscillations and auto-waves. Classification of self-oscillatory systems.- 16 Examples of self-oscillatory systems with lumped parameters. I.- 17 Examples of self-oscillatory systems with lumped parameters. II.- 18 Examples of self-oscillatory systems with high frequency power sources.- 19 Examples of self-oscillatory systems with time delay.- 20 Examples of continuous self-oscillatory systems with lumped active elements.- 21 Examples of self-oscillatory systems with distributed active elements.- 22 Periodic actions on self-oscillatory systems. Synchronization and chaotization of self-oscillations.- 23 Interaction between self-oscillatory systems.- 24 Examples of auto-waves and dissipative structures.- 25 Convective structures and self-oscillations in fluid. The onset of turbulence.- 26 Hydrodynamic and acoustic waves in subsonic jet and separated flows.- Appendix A Approximate methods for solving linear differential equations with slowly varying parameters.- A.1 JWKB Method.- A.2 Asymptotic method.- A.3 The LiouvilleGreen transformation.- A.4 The Langer transformation.- Appendix B The Whitham method and the stability of periodic running waves for the KleinGordon equation.