ISBN: 3540410015
TITLE: Regular and Chaotic Oscillations
AUTHOR: Landa
TOC:

1 Introduction 1
1.1 The importance of oscillation theory for engineering mechanics 1
1.2 Classification of dynamical systems. systems with conservation of Phase volume and dissipative systems 2
1.3 Different types of mathematical models and their functions in studies of concrete systems 5
1.4 Phase space of autonomous dynamical systems and the number of degrees of freedom 6
1.5 The subject matter of the book 7
2. The main analytical methods of studies of nonlinear oscillations in near-conservative systems 9
2.1 The van der Pol method 10
2.2 The asymptotic Krylov-Bogolyubov method 12
2.3 The averaging method 16
2.4 The averaging method in systems incorporating fast and slow variables 19
2.5 The Whitham method 21
Part 1. OSCILLATIONS IN AUTONOMOUS DYNAMICAL SYSTEMS
3. General properties of autonomous dynamical systems 27
3.1 Phase space of autonomous dynamical systems and its structure. Singular Points and limit sets 27
3.1.1 Singular Points and their classification 27
3.1.2 Stability criterion of singular points 30
3.2 Attractors and repellers 32
3.3 The stability of limit cycles and their classification 33
3.4 Strange attractors: stochastic and chaotic attractors 35
3.4.1 Quantitative characteristics of attractors 36
3.4.2 Reconstruction of attractors from experimental data 41
3.5 Poincar cutting surface and point maps 42
3.6 Some routes for the loss of stability of simple attractors and the appearance of strange attractors 44
3.7 Integrable and nonintegrable systems. Action-angle variables 45
4. Examples of natural oscillations in systems with one degree of freedom 49
4.1 Oscillator with nonlinear restering force 49
4.1.1 Pendulum oscillations 49
4.1.2 Oscillations of a pendulum placed between the opposite poles of a magnet 51
4.1.3 Oscillations described by Duffing equations 52
4.1.4 Oscillations of a material point in a forte field with the Toda potential 54
4.2 Oscillations of a bubble in fluid 56
4.3 Oscillations of species populations described by the Lotka-Volterra equations 58
4.4 Natural oscillations in a System with slowly time-varying natural frequency 61
5. Natural oscillations in systems with many degrees of freedom. Normal oscillations 63
5.1 Normal oscillations in linear conservative systems 63
5.2 Normal oscillations in nonlinear conservative systems 64
5.3 Examples of normal oscillations in linear and nonlinear conservative systems 65
5.3.1 Two coupled linear oscillators with gyroscopic forces 65
5.3.2 Examples of normal oscillations in two coupled nonlinear oscillators 66
5.3.3 An example of normal oscillations in three coupled nonlinear oscillators 67
5.3.4 Normal oscillations in linear homogeneous and periodically inhomogeneous chains 71
5.3.5 Examples of natural oscillations in nonlinear homogeneous chains 78
5.4 Stochasticity in Hamiltonian systems close to integrable ones 84
5.4.1 The ring Toda chain and the Henon-Heiles System 84
5.4.2 Stochastization of oscillations in the Yang-Mills equations 87
6. Self-oscillatory systems with one degree of freedom 89
6.1 The van der Pol, Bayleigh and Bautin equations 89
6.1.1 The Kaidanovsky-Khaikin frictional generator and the Froude pendulum 92 6.2 Soft and hard excitation of self-oscillations 93
6.3 Truncated equations for the oscillation amplitude and phase 95
6.3.1 Quasi-linear systems 95
6.3.2 Transient processes in the van der Pol generator 98
6.3.3 Essentially nonlinear quasi-conservative systems 98
6.4 The Rayleigh relaxation generator 100
6.5 Clock movement mechanisms and the Neimark pendulum. The energetic criterion of chaotization of self-oscillations 102
7. Self-oscillatory systems with one and a half degrees of freedom lO7
7.1 Self-oscillatory systems with inertial excitation 107
7.1.1 The model equations of self-oscillatory systems with inertial excitation 107
7.1.2 Examples of self-oscillatory systems with inertial excitation 110
7.2 Self-oscillatory systems with inertial nonlinearity 127
7.3 Some other systems with one and a half degrees of freedom 132
7.3.1 The Rssler equations 132
7.3.2 A three-dimensional model of an immune reaction illustrating the oscillatory course of some chronic diseases 134
8. Examples of self-oscillatory systems with two or more degrees of freedom
137
8.1 Generator with an additional circuit 137
8.2 A lumped model of bending-torsion flutter of an aircraft wing 141
8.3 A model of the vocal source 145
8.4 The lumped model of a 'Singing' flame 152
8.5 A self-oscillatory System based on a ring Toda chain 155
9. Synchronization and chaotization of self-oscillatory systems by an extemal harmonic forte 161
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 161
9.1.1 The main resonance 162
9.1.2 Resonances of the nth kind 167
9.2 Synchronization of a generator with hard excitation. Asynchronous excitation of self-oscillations 176
9.2.1 Asynchronous excitation of self-oscillations 179
9.3 Synchronization of the van der Pol generator with modulated natural frequency 180
9.4 Synchronization of periodic oscillations in systems with inertial nonlinearity 188
9.5 Chaotization of periodic self-oscillations by an external forte 193
9.6 Synchronization of chaotic self-oscillations. The synchronization threshold and its relation to the quantitative characteristics of the attractor 195
9.7 Synchronization of vortex formation in the case of transverse flow around a vibrated cylinder 196
9.8 Synchronization of relaxation self-oscillations 199
10. Interaction of two self-oscillatory systems. Synchronization and chaotization of self-oscillations 205
10.1 Mutual synchronization of periodic self-oscillations with close frequencies 205
10.1.1 The case of weak linear coupling 206
10.1.2 The case of strong linear coupling 213
10.2 Mutual synchronization of self-oscillations with multiple frequencies 216
10.3 Parametric synchronization of two generators with different frequencies 219
10.4 Chaotization of self-oscillations in two coupled generators 220
10.5 Interaction of generators of periodic and chaotic oscillations 223
10.6 Interaction of generators of chaotic oscillations 225
10.7 Mutual synchronization of two relaxation generators 231
10.7.1 Mutual synchronization of two coupled relaxation generators of triangular oscillations 231
10.7.2 Mutual synchronization of two Rayleigh relaxation generators 233
11. Interaction of three or more self-oscillatory systems 237
11.1 Mutual synchronization of three generators 237
11.1.1 The case of close frequencies 237
11.1.2 The case of close differentes of the frequencies of neighboring generators 243
11.2 Synchronization of N coupled generators with close frequencies 244
11.2.1 Synchronization of N coupled van der Pol generators 244
11.2.2 Synchronization of pendulum clocks suspended from a common beam 246
11.3 Synchronization and chaotization of self-oscillations in chains of coupled generators 248
11.3.1 Synchronization of N van der Pol generators coupled in a chain 248
11.3.2 Synchronization and chaotization of self-oscillations in a chain of N coupled van der Pol-Duffing generators 249
11.3.3 Synchronization of chaotic oscillations in a chain of generators with inertial nonlinearity 250
Part 11. OSCILLATIONS IN NONAUTONOMOUS SYSTEMS
12. Oscillations of nonlinear systems excited by external periodic forces 255
12.1 A periodically driven nonlinear oscillator 255
12.1.1 The main resonance 257
12.1.2 Subharmonic resonances 260
12.1.3 Superharmonic resonances 262
12.2 Oscillations excited by an external forte with a slowly time varying frequency 263
12.3 Chaotic regimes in periodically driven nonlinear oscillators 266
12.3.1 Chaotic regimes in the Duffing oscillator 267
12.3.2 Chaotic oscillations of a gas bubble in liquid under the action of a sound field 267
12.3.3 Chaotic oscillations in the Vallis model 268
12.4 Two coupled harmonically driven nonlinear oscillators 269
12.4.1 The main resonance 270
12.4.2 The combination resonance 275
12.5 Electro-mechanical Vibrators and capacitative sensors of small displacements 283
13. Parametric excitation of oscillations 289
13.1 Parametrically excited nonlinear oscillators 289
13.1.1 Slightly nonlinear oscillator with small damping and small harmonic parametric action 289
13.2 Chaotization of a parametrically excited nonlinear oscillator 292
13.3 Parametric excitation of pendulum oscillations by noise 294
13.3.1 The results of a numerical simulation of the oscillations of a pendulum with a randomly vibrated Suspension axis 298
13.3.2 On-Off intermittency 299
13.3.3 Correlation dimension 302
13.3.4 Power spectra 302
13.3.5 The Rytov-Dimentberg criterion 303
13.4 Parametric resonance in a System of two coupled oscillators 305
13.5 Simultaneous forced and parametric excitation of an oscillator 312
13.5.1 Parametric amplifier 312
13.5.2 Regular and chaotic oscillations in a model of child-hood infections 314
14. Changes in the dynamical behavior of nonlinear systems induced by high-frequency vibration or by noise 323
14.1 The appearance and disappearance of attractors and repellers induced by high-frequency vibration or noise 323
14.2 Vibrational transport and electrical rectification 331
14.2.1 Vibrational transport 332
14.2.2 Rectification of fluctuations 335
14.3 Noise-induced transport of Brownian particles (stochastic ratchets) 336
14.3.1 Noise-induced transport of light Brownian particles in a viscous medium with a saw-tooth potential 337
14.3.2 The effect of the particle mass 344
14.4 Stochastic and vibrational resonances: similarities and distinctions 359
14.4.1 Stochastic resonance in an overdamped oscillator 362
14.4.2 Vibrational resonance in an overdamped oscillator 366
14.4.3 Stochastic and vibrational resonances in a weakly damped bistable oscillator. Control of resonance 367
A. Derivation of the approximate equation for the one-dimensional probability density 373
References 376
Index 393
END
