Nonlinear Oscillations and Waves in Dynamical Systems

Transcription

1 Nonlinear Oscillations and Waves in Dynamical Systems by P. S. Landa Department of Physics, Moscow State University, Moscow, Russia KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

2 Contents Preface xiii Introduction 1 1 The purpose and subject matter of the book 1 2 The definition and significance of the theory of oscillations and waves. The subject area of its investigations. The history of the creation and development of this theory. The relation between the theory of oscillations and waves and the problems of synergetics 3 Part I BASIC NOTIONS AND DEFINITIONS 7 Chapter 1 Dynamical systems. Phase space. Stochastic and chaotic systems. The number of degrees of freedom Definition of a dynamical system and its phase space Classification of dynamical systems. The concept of energy Integrable and non-integrable systems. Action-angle variables Systems with slowly time varying parameters. Adiabatic invariants Dissipative systems. Amplifiers and generators 17 Chapter 2 Hamiltonian systems close to integrable. Appearance of stochastic motions in Hamiltonian systems The content of the Kolmogorov-Arnold-Moser theory The Henon-Heiles system 20 Chapter 3 Attractors and repellers. Reconstruction of attractors from an experimental time series. Quantitative characteristics of attractors Simple and complex attractors and repellers. Stochastic and chaotic attractors Reconstruction of attractors from an experimental time series Quantitative characteristics of attractors 25 Chapter 4 Natural and forced oscillations and waves. Self-oscillations and auto-waves Natural and forced oscillations and waves 28

3 4.2 Self-oscillations and auto-waves 30 Part II BASIC DYNAMICAL MODELS OF THE THEORY OF OSCILLATIONS AND WAVES 33 Chapter 5 Conservative systems Harmonic oscillator Anharmonic oscillator The Lotka-Volterra system ('prey-predator' model) Chains of nonlinear oscillators. The Toda and Fermi-Pasta-Ulam chains The wave equation. The Klein-Gordon and sine-gordon equations. The Born-Infeld equation The equation of simple (Riemann) waves The Boussinesq and Korteweg-de Vries equations The Whitham and Rudenko equations The Khokhlov-Zabolotskaya, cubic Schrodinger, Ginsburg-Landau, and Hirota equations Some discrete models of conservative systems 56 Chapter 6 Non-conservative Hamiltonian systems and dissipative systems Non-linear damped oscillator with an external force The Burgers and Burgers-Korteweg-de Vries equations The van der Pol, Rayleigh, and Bautin equations The equations of systems with inertial excitation and inertial non-linearity The Lorenz, Rossler, and Chua equations A model of an active string Models for locally excited media(the equation for a. kink wave, the Fitz Hugh-Nagumo and Turing equations) The Kuramoto-Sivashinsky equation The Feigenbaum and Zisook maps 67 Part III NATURAL (FREE) OSCILLATIONS AND WAVES IN LINEAR AND NON-LINEAR SYSTEMS 69 Chapter 7 Natural oscillations of non-linear oscillators Pendulum oscillations Oscillations described by the Duffing equation Oscillations of a material point in a force field with the Toda potential Oscillations of a bubble in fluid Oscillations of species strength described by the Lotka-Volterra equations 81

4 Vll 7.6 Oscillations in a system with slowly time varying natural frequency. 81 Chapter 8 Natural oscillations in systems of coupled oscillators Linear conservative systems. Normal oscillations Oscillations in linear homogeneous and periodically inhomogeneous chains Normal oscillations in non-linear conservative systems Oscillations in non-linear homogeneous chains Oscillations of coupled non-linear damped oscillators. Homoclinic structures. A model of acoustic emission 102 Chapter 9 Natural waves in bounded and unbounded continuous media. Solitons Normally and anomalously dispersive linear waves. Ionization waves in plasmas. Planetary waves in ocean (Rossby waves and solitons) Non-linear waves described by the Born-Infeld equation. Solitons of the Klein-Gordon and sine-gordon equations. Interaction between solitons Ill 9.3 Simple, saw-tooth and shock waves Solitons of the Korteweg-de Vries equation Stationary waves described by the Burgers-Korteweg-de Vries equationl Solitons of the Boussinesq equation Solitons of the cubic Schrodinger and Ginsburg-Landau equations Natural waves in slightly inhomogeneous and slightly non-stationary media. The wave action as an adiabatic invariant Natural waves in periodically stratified media 133 Part IV FORCED OSCILLATIONS AND WAVES IN PASSIVE SYSTEMS 137 Chapter 10 Oscillations of a non-linear oscillator excited by an external force Periodically driven non-linear oscillators. The main, subharmonic and superharmonic resonances The main resonance Subharmonic resonances Superharmonic resonances Chaotic oscillations of non-linear systems under periodic external actions Chaotic oscillations described by the Duffing equation Chaotic oscillations of a gas bubble in liquid under the action of a sound field Chaotic oscillations in the Vallis model for non-linear interaction between ocean and atmosphere 149

5 Vlll 10.3 Oscillations excited by external force with a slowly time varying frequency 152 Chapter 11 Oscillations of coupled non-linear oscillators excited by an external periodic force The main resonance in a system of two coupled harmonically excited non-linear oscillators Combination resonances in two coupled harmonically driven non-linear oscillators Driven oscillations in linear homogeneous and periodically inhomogeneous chains caused by a harmonic force applied to the input of the chain Forced oscillations in non-linear homogeneous and periodically inhomogeneous chains caused by a harmonic force applied to the input of the chain. Excitation of the second harmonic and decay instability Driven vibration of a string excited by a distributed external harmonic force 184 Chapter 12 Parametric oscillations Parametrically excited non-linear oscillator Slightly non-linear oscillator with small damping and small harmonic action High frequency parametric action upon a pendulum. Stabilization of the upper equilibrium position as an induced phase transition Chaotization of a parametrically excited non-linear oscillator. Regular and chaotic oscillations in a model of childhood infections accounting for periodic seasonal change of the contact rate Parametric resonances in a system of two coupled oscillators Simultaneous forced and parametric excitation of a linear oscillator. Parametric amplifier 199 Chapter 13 Waves in semibounded media excited by perturbations applied to their boundaries One-dimensional waves in non-linear homogeneous non-dispersive media. Shock and saw-tooth waves One-dimensional waves in non-linear homogeneous slightly dispersive media described by the Korteweg-de Vries equation One-dimensional waves in non-linear highly dispersive media Non-linear wave bundles in dispersive media Self-focusing and self-defocusing of wave bundles Compression and expantion of pulses in non-linear dispersive media Non-linear wave bundles in non-dispersive media. Approximate solutions of the Khokhlov-Zabolotskaya equation Waves in slightly inhomogeneous media 220

6 13.7 Waves in periodically inhomogeneous media 223 Part V OSCILLATIONS AND WAVES IN ACTIVE SYSTEMS. SELF-OSCILLATIONS AND AUTO-WAVES 225 Chapter 14 Forced oscillations and waves in active non-self-oscillatory systems. Turbulence. Burst instability. Excitation of waves with negative energy Amplifiers with lumped parameters Continuous semibounded media with convective instability Excitation of turbulence in non-closed fluid flows. The Klimontovich criterion of motion ordering One-dimensional waves in active non-linear media. Burst instability Waves with negative energy and instability caused by them 235 Chapter 15 Mechanisms of excitation and amplitude limitation of self-oscillations and auto-waves. Classification of self-oscillatory systems Mechanisms of excitation and amplitude limitation of self-oscillations in the simplest systems. Soft and hard excitation of self-oscillations Mechanisms of the excitation of self-oscillations in systems with high frequency power sources Mechanisms of excitation of self-oscillations in continuous systems. Absolute instability as a mechanism of excitation of auto-waves Quasi-harmonic and relaxation self-oscillatory systems. Stochastic and chaotic systems Possible routes for loss of stability of regular motions and the appearance of chaos and stochasticity The Feigenbaum scenario The transition to chaos via fusion of a stable limit cycle with an unstable one and the subsequent disappearance of both of these cycles The transition to chaos via destruction of a two-dimensional torus The Ruelle-Takens scenario 245 Chapter 16 Examples of self-oscillatory systems with lumped parameters. I Electronic generator. The van der Pol and Rayleigh equations The Kaidanovsky-Khaikin frictional generator and the Froude pendulum The Bonhoeffer-van der Pol oscillator A model of glycolysis and a lumped version of the 'brusselator' A lumped model of the Buravtsev oscillator 256

7 16.6 Clock movement mechanisms and the Neimark pendulum. The energetic criterion of self-oscillation chaotization Self-oscillatory models for species interaction based on the Lotka-Volterra equations Systems with inertial non-linearity The Pikovsky model Systems with inertial excitation The Helmholtz resonator with non-uniformly heated walls A heated wire with a weight at its centre A modified 'brusselator' Self-oscillations of an air cushioned body 277 Chapter 17 Examples of self-oscillatory systems with lumped parameters. II The Rossler and Chua systems A three-dimensional model of an immune reaction illustrating an -oscillatory course of some chronic diseases. The 'oregonator' model The simplest model of the economic progress of human society Models of the vocal source A lumped model of the 'singing' flame 303 Chapter 18 Examples of self-oscillatory systems with high frequency power sources The Duboshinsky pendulum, a 'gravitational machine', and the Andreev hammer The Bethenod pendulum, the Papaleksi effect, and the Rytov device Electro-mechanical vibrators. Capacitance sensors of small displacements 317 Chapter 19 Examples of self-oscillatory systems with time delay Biological controlled systems Models of respiration control The Mackey-Glass model of the process of regeneration of white blood corpuscles (neutrophils) Models of the control of upright human posture The van der Pol-Duffing generator with additional delayed feedback as a model of Doppler's autodyne A ring optical cavity with an external field (the Ikeda system) Chapter 20 Examples of continuous self-oscillatory systems with lumped active elements The Vitt system. Competition and synchronization of modes The Rijke phenomenon A distributed model of the 'singing' flame 351

8 XI Chapter 21 Examples of self-oscillatory systems with distributed active elements Lasers. Competition, synchronization and chaotization of modes. Optical auto-solitons The Gann generators Ionization waves (striations) in low temperature plasmas Inert gases Molecular gases A model of the generation of. Korotkov's sounds Self-oscillations of a bounded membrane resulting from excitation of waves with negative energy 393 Chapter 22 Periodic actions on self-oscillatory systems. Synchronization and chaotization of self-oscillations Synchronization of periodic self-oscillations by an external force in the van der Pol-Duffing generator. Two mechanisms of synchronization. Synchronization as a non-equilibrium phase transition Synchronization of periodic oscillations in a generator with inertial non-linearity and in more complicated systems Synchronization of a van der Pol generator with a modulated natural frequency Asynchronous quenching and asynchronous excitation of periodic self-oscillations Chaotization of periodic self-oscillations by a periodic external force Synchronization of chaotic self-oscillations. The synchronization threshold and its relation to the quantitative characteristics of the attractor 412 Chapter 23 Interaction between self-oscillatory systems Mutual synchronization of two generators of periodic oscillations Mutual synchronization of three and more coupled generators of periodic oscillations Chaotization of self-oscillations in systems of coupled generators Interaction between generators of periodic and chaotic oscillations Interaction between generators of chaotic oscillations. The notion of synchronization 426 Chapter 24 Examples of auto-waves and dissipative structures Auto-waves of burning. A model of a kink wave Auto-waves in the Fitz Hugh-Nagumo model Auto-waves in a distributed version of the brusselator and in some other models of biological, chemical and ecological systems Auto-waves described by the Kuramoto-Sivashinsky equation and the generalized Kuramoto-Sivashinsky equation 440

9 Xll Chapter 25 Convective structures and self-oscillations in fluid. The onset of turbulence Rayleigh-Taylor instability and the initial stage of the excitation of thermo-convection in a plane layer Thermo-convection in a toroidal tube. The Lorenz equations The initial stage of excitation of bio-convection Onset of turbulence in theflowbetween two coaxial rotating cylinders. Taylor vortices 456 Chapter 26 Hydrodynamic and acoustic waves in subsonic jet and separated flows The Kelvin-Helmholtz instability Subsonic free jets Sound excitation by an impinging jet. Excitation of edgetones Self-oscillations in open jet return circuit wind tunnels The von Karman vortex wake, Aeolian tones and stalling flutter Appendix A Approximate methods for solving linear differential equations with slowly varying parameters 489 A.I JWKB Method 489 A.2 Asymptotic method 490 A.3 The Liouville-Green transformation 491 A.4 The Langer transformation 492 Appendix B The Whitham method and the stability of periodic running waves for the Klein Gordon equation 494 Bibliography 499 Index 535