Regular and Chaotic Oscillations

Abstract

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 Poincare 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 Rossler 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 frequencies.- 11.1.2 The case of close differences of the frequencies of neighboring generators.- 11.2 Synchronization of N coupled generators with close frequencies.- 11.2.1 Synchronization of N coupled van der Pol generators.- 11.2.2 Synchronization of pendulum clocks suspended from a common beam.- 11.3 Synchronization and chaotization of self-oscillations in chains of coupled generators.- 11.3.1 Synchronization of N van der Pol generators coupled in a chain.- 11.3.2 Synchronization and chaotization of self-oscillations in a chain of N coupled van der Pol-Duffing generators.- 11.3.3 Synchronization of chaotic oscillations in a chain of generators with inertial nonlinearity.- II. Oscillations in Nonautonomous Systems.- 12. Oscillations of nonlinear systems excited by external periodic forces.- 12.1 A periodically driven nonlinear oscillator.- 12.1.1 The main resonance.- 12.1.2 Subharmonic resonances.- 12.1.3 Superharmonic resonances.- 12.2 Oscillations excited by an external force with a slowly time-varying frequency.- 12.3 Chaotic regimes in periodically driven nonlinear oscillators.- 12.3.1 Chaotic regimes in the Duffing oscillator.- 12.3.2 Chaotic oscillations of a gas bubble in liquid under the action of a sound field.- 12.3.3 Chaotic oscillations in the Vallis model.- 12.4 Two coupled harmonically driven nonlinear oscillators.- 12.4.1 The main resonance.- 12.4.2 The combination resonance.- 12.5 Electro-mechanical vibrators and capacitative sensors of small displacements.- 13. Parametric excitation of oscillations.- 13.1 Parametrically excited nonlinear oscillators.- 13.1.1 Slightly nonlinear oscillator with small damping and small harmonic parametric action.- 13.2 Chaotization of a parametrically excited nonlinear oscillator.- 13.3 Parametric excitation of pendulum oscillations by noise.- 13.3.1 The results of a numerical simulation of the oscillations of a pendulum with a randomly vibrated suspension axis.- 13.3.2 On-off intermittency.- 13.3.3 Correlation dimension.- 13.3.4 Power spectra.- 13.3.5 The Rytov-Dimentberg criterion.- 13.4 Parametric resonance in a system of two coupled oscillators.- 13.5 Simultaneous forced and parametric excitation of an oscillator.- 13.5.1 Parametric amplifier.- 13.5.2 Regular and chaotic oscillations in a model of childhood infections.- 14. Changes in the dynamical behavior of nonlinear systems induced by high-frequency vibration or by noise.- 14.1 The appearance and disappearance of attractors and repellers induced by high-frequency vibration or noise.- 14.2 Vibrational transport and electrical rectification.- 14.2.1 Vibrational transport.- 14.2.2 Rectification of fluctuations.- 14.3 Noise-induced transport of Brownian particles (stochastic ratchets).- 14.3.1 Noise-induced transport of light Brownian particles in a viscous medium with a saw-tooth potential.- 14.3.2 The effect of the particle mass.- 14.4 Stochastic and vibrational resonances: similarities and distinctions.- 14.4.1 Stochastic resonance in an overdamped oscillator.- 14.4.2 Vibrational resonance in an overdamped oscillator.- 14.4.3 Stochastic and vibrational resonances in a weakly damped bistable oscillator. Control of resonance.- A. Derivation of the approximate equation for the one-dimensional probability density.- References.