Regular and Chaotic Oscillations

In the present book the modern theory of non-linear oscilla- tions both regular and chaotic, is set out, primarily, as applied to mechanical problems. The material is presented in a non-traditional manner with emphasizing of the new results of the theory otained 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 sy- stems, the influence of weak random vibration on modificati- on of characteristics and behavior of the non-linear systems etc. One of the purposes of the book is to convince the rea- ders of the necessity of a thorough study of this theory and to show that it can be very useful in engineering investiga- tions. The primary readers for this book are researchers working with different oscillatory processes, and both un- der-graduate and post-graduate students interested in a deep study of the general laws and applications of the theory of nonlinear oscillations. The instructors can find here new materials for lecturess and optinal courses.

Klappentext

This text maps out the modern theory of non-linear oscillations. The material is presented in a non-traditional manner and emphasises the new results of the theory - obtained partially by the author, who is one of the leading experts in the area. Among the topics are: synchronization and chaotization of self-oscillatory systems and the influence of weak random vibration on modification of characteristics and behaviour of the non-linear systems.

Inhalt

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.- 3.1 Phase space of autonomous dynamical systems and its structure. Singular points and limit sets.- 3.1.1 Singular points and their classification.- 3.1.2 Stability criterion of singular points.- 3.2 Attractors and repellers.- 3.3 The stability of limit cycles and their classification.- 3.4 Strange attractors: stochastic and chaotic attractors.- 3.4.1 Quantitative characteristics of attractors.- 3.4.2 Reconstruction of attractors from experimental data.- 3.5 Poincaré cutting surface and point maps.- 3.6 Some routes for the loss of stability of simple attractors and the appearance of strange attractors.- 3.7 Integrable and nonintegrable systems. Action-angle variables.- 4. Examples of natural oscillations in systems with one degree of freedom.- 4.1 Oscillator with nonlinear restoring force.- 4.1.1 Pendulum oscillations.- 4.1.2 Oscillations of a pendulum placed between the opposite poles of a magnet.- 4.1.3 Oscillations described by Duffing equations.- 4.1.4 Oscillations of a material point in a force field with the Toda potential.- 4.2 Oscillations of a bubble in fluid.- 4.3 Oscillations of species populations described by the Lotka-Volterra equations.- 4.4 Natural oscillations in a system with slowly time-varying natural frequency.- 5. Natural oscillations in systems with many degrees of freedom. Normal oscillations.- 5.1 Normal oscillations in linear conservative systems.- 5.2 Normal oscillations in nonlinear conservative systems.- 5.3 Examples of normal oscillations in linear and nonlinear conservative systems.- 5.3.1 Two coupled linear oscillators with gyroscopic forces.- 5.3.2 Examples of normal oscillations in two coupled nonlinear oscillators.- 5.3.3 An example of normal oscillations in three coupled nonlinear oscillators.- 5.3.4 Normal oscillations in linear homogeneous and periodically inhomogeneous chains.- 5.3.5 Examples of natural oscillations in nonlinear homogeneous chains.- 5.4 Stochasticity in Hamiltonian systems close to integrable ones.- 5.4.1 The ring Toda chain and the Henon-Heiles system.- 5.4.2 Stochastization of oscillations in the Yang-Mills equations.- 6. Self-oscillatory systems with one degree of freedom.- 6.1 The van der Pol, Rayleigh and Bautin equations.- 6.1.1 The Kaidanovsky-Khaikin frictional generator and the Froude pendulum.- 6.2 Soft and hard excitation of self-oscillations.- 6.3 Truncated equations for the oscillation amplitude and phase.- 6.3.1 Quasi-linear systems.- 6.3.2 Transient processes in the van der Pol generator.- 6.3.3 Essentially nonlinear quasi-conservative systems.- 6.4 The Rayleigh relaxation generator.- 6.5 Clock movement mechanisms and the Neimark pendulum. The energetic criterion of chaotization of self-oscillations.- 7. Self-oscillatory systems with one and a half degrees of freedom.- 7.1 Self-oscillatory systems with inertial excitation.- 7.1.1 The model equations of self-oscillatory systems with inertial excitation.- 7.1.2 Examples of self-oscillatory systems with inertial excitation.- 7.2 Self-oscillatory systems with inertial nonlinearity.- 7.3 Some other systems with one and a half degrees of freedom.- 7.3.1 The Rössler equations.- 7.3.2 A three-dimensional model of an immune reaction illustrating the oscillatory course of some chronic diseases.- 8. Examples of self-oscillatory systems with two or more degrees of freedom.- 8.1 Generator with an additional circuit.- 8.2 A lumped model of bending-torsion flutter of an aircraft wing.- 8.3 A model of the vocal source.- 8.4 The lumped model of a 'singing' flame.- 8.5 A self-oscillatory system based on a ring Toda chain.- 9. Synchronization and chaotization of self-oscillatory systems by an external harmonic force.- 9.1 Synchronization of self-oscillations by an external periodic force in a system with one degree of freedom with soft excitation. Two mechanisms of synchronization.- 9.1.1 The main resonance.- 9.1.2 Resonances of the nth kind.- 9.2 Synchronization of a generator with hard excitation. Asynchronous excitation of self-oscillations.- 9.2.1 Asynchronous excitation of self-oscillations.- 9.3 Synchronization of the van der Pol generator with modulated natural frequency.- 9.4 Synchronization of periodic oscillations in systems with inertial nonlinearity.- 9.5 Chaotization of periodic self-oscillations by an external force.- 9.6 Synchronization of chaotic self-oscillations. The synchronization threshold and its relation to the quantitative characteristics of the attractor.- 9.7 Synchronization of vortex formation in the case of transverse flow around a vibrated cylinder.- 9.8 Synchronization of relaxation self-oscillations.- 10. Interaction of two self-oscillatory systems. Synchronization and chaotization of self-oscillations.- 10.1 Mutual synchronization of periodic self-oscillations with close frequencies.- 10.1.1 The case of weak linear coupling.- 10.1.2 The case of strong linear coupling.- 10.2 Mutual synchronization of self-oscillations with multiple frequencies.- 10.3 Parametric synchronization of two generators with different frequencies.- 10.4 Chaotization of self-oscillations in two coupled generators.- 10.5 Interaction of generators of periodic and chaotic oscillations.- 10.6 Interaction of generators of chaotic oscillations.- 10.7 Mutual synchronization of two relaxation generators.- 10.7.1 Mutual synchronization of two coupled relaxation generators of triangular oscillations.- 10.7.2 Mutual synchronization of two Rayleigh relaxation generators.- 11. Interaction of three or more self-oscillatory systems.- 11.1 Mutual synchronization of three generators.- 11.1.1 The case of close frequ…