Nonlinear Oscillations And Waves In Dynamical Systems
Nonlinear Oscillationsand Waves inDynamical SystemsbyP. S. LandaDepartment of Physics,Moscow State University,Moscow, RussiaKLUWER ACADEMIC PUBLISHERSDORDRECHT / BOSTON / LONDON
ContentsPrefacexiiiIntroduction1The purpose and subject matter of the book2The definition and significance of the theory of oscillations and waves.The subject area of its investigations. The history of the creationand development of this theory. The relation between the theory ofoscillations and waves and the problems of synergetics11Part I7BASIC NOTIONS AND DEFINITIONS3Chapter 1 Dynamical systems. Phase space. Stochastic and chaoticsystems. The number of degrees of freedom91.1 Definition of a dynamical system and its phase space91.2 Classification of dynamical systems. The concept of energy101.3 Integrable and non-integrable systems. Action-angle variables . . . . 131.4 Systems with slowly time varying parameters. Adiabatic invariants . 161.5 Dissipative systems. Amplifiers and generators17Chapter 2 Hamiltonian systems close to integrable. Appearance ofstochastic motions in Hamiltonian systems192.1 The content of the Kolmogorov-Arnold-Moser theory192.2 The Henon-Heiles system20Chapter 3 Attractors and repellers. Reconstruction of attractorsfrom an experimental time series. Quantitative characteristics ofattractors223.1 Simple and complex attractors and repellers. Stochastic and chaoticattractors223.2 Reconstruction of attractors from an experimental time series . . . . 243.3 Quantitative characteristics of attractors25Chapter 4 Natural and forced oscillations and waves. Self-oscillationsand auto-waves284.1Natural and forced oscillations and waves28
4.2Self-oscillations and auto-waves30Part II BASIC DYNAMICAL MODELS OF THETHEORY OF OSCILLATIONS AND WAVES33Chapter 5 Conservative systems5.1 Harmonic oscillator5.2 Anharmonic oscillator5.3 The Lotka-Volterra system ('prey-predator' model)5.4 Chains of nonlinear oscillators. The Toda and Fermi-Pasta-Ulamchains5.5 The wave equation. The Klein-Gordon and sine-Gordon equations.The Born-Infeld equation5.6 The equation of simple (Riemann) waves5.7 The Boussinesq and Korteweg-de Vries equations5.8 The Whitham and Rudenko equations5.9 The Khokhlov-Zabolotskaya, cubic Schrodinger, Ginsburg-Landau,and Hirota equations5.10 Some discrete models of conservative systemsChapter 6 Non-conservative Hamiltonian systems and dissipativesystems6.1 Non-linear damped oscillator with an external force6.2 The Burgers and Burgers-Korteweg-de Vries equations6.3 The van der Pol, Rayleigh, and Bautin equations6.4 The equations of systems with inertial excitation and inertialnon-linearity6.5 The Lorenz, Rossler, and Chua equations6.6 A model of an active string6.7 Models for locally excited media(the equation for a. kink wave, theFitz Hugh-Nagumo and Turing equations)6.8 The Kuramoto-Sivashinsky equation6.9 The Feigenbaum and Zisook maps353536363739414350515658585962626365656667Part III NATURAL (FREE) OSCILLATIONS ANDWAVES IN LINEAR AND NON-LINEAR SYSTEMS 69Chapter 7 Natural oscillations of non-linear oscillators7.1 Pendulum oscillations7.2 Oscillations described by the Duffing equation7.3 Oscillations of a material point in a force field with the Toda potential7.4 Oscillations of a bubble influid7.5 Oscillations of species strength described by the Lotka-Volterraequations717172757781
Vll7.6Oscillations in a system with slowly time varying natural frequencyChapter 8 Natural oscillations in systems of coupled oscillators8.1 Linear conservative systems. Normal oscillations8.2 Oscillations in linear homogeneous and periodically inhomogeneouschains8.3 Normal oscillations in non-linear conservative systems8.4 Oscillations in non-linear homogeneous chains8.5 Oscillations of coupled non-linear damped oscillators. Homoclinicstructures. A model of acoustic emission. 818585879399102Chapter 9 Natural waves in bounded and unbounded continuousmedia. Solitons1069.1 Normally and anomalously dispersive linear waves. Ionization wavesin plasmas. Planetary waves in ocean (Rossby waves and solitons) . . 1069.2 Non-linear waves described by the Born-Infeld equation. Solitons ofthe Klein-Gordon and sine-Gordon equations. Interaction betweensolitonsIll9.3 Simple, saw-tooth and shock waves1169.4 Solitons of the Korteweg-de Vries equation1219.5 Stationary waves described by the Burgers-Korteweg-de Vries equationl269.6 Solitons of the Boussinesq equation1269.7 Solitons of the cubic Schrodinger and Ginsburg-Landau equations . . 1279.8 Natural waves in slightly inhomogeneous and slightly non-stationarymedia. The wave action as an adiabatic invariant1299.9 Natural waves in periodically stratified media133Part IV FORCED OSCILLATIONS AND WAVES INPASSIVE SYSTEMS137Chapter 10 Oscillations of a non-linear oscillator excited by anexternal force13910.1 Periodically driven non-linear oscillators. The main, subharmonicand superharmonic resonances13910.1.1 The main resonance14110.1.2 Subharmonic resonances14410.1.3 Superharmonic resonances14610.2 Chaotic oscillations of non-linear systems under periodic externalactions14710.2.1 Chaotic oscillations described by the Duffing equation14810.2.2 Chaotic oscillations of a gas bubble in liquid under the actionof a soundfield14910.2.3 Chaotic oscillations in the Vallis model for non-linearinteraction between ocean and atmosphere149
Vlll10.3 Oscillations excited by external force with a slowly time varyingfrequency152Chapter 11 Oscillations of coupled non-linear oscillators excited by anexternal periodic force15611.1 The main resonance in a system of two coupled harmonically excitednon-linear oscillators15611.2 Combination resonances in two coupled harmonically drivennon-linear oscillators16111.3 Driven oscillations in linear homogeneous and periodicallyinhomogeneous chains caused by a harmonic force applied to the inputof the chain16711.4 Forced oscillations in non-linear homogeneous and periodicallyinhomogeneous chains caused by a harmonic force applied to the inputof the chain. Excitation of the second harmonic and decay instability 17311.5 Driven vibration of a string excited by a distributed external harmonicforce184Chapter 12 Parametric oscillations12.1 Parametrically excited non-linear oscillator12.1.1 Slightly non-linear oscillator with small damping and smallharmonic action12.1.2 High frequency parametric action upon a pendulum.Stabilization of the upper equilibrium position as an inducedphase transition12.2 Chaotization of a parametrically excited non-linear oscillator.Regular and chaotic oscillations in a model of childhood infectionsaccounting for periodic seasonal change of the contact rate12.3 Parametric resonances in a system of two coupled oscillators12.4 Simultaneous forced and parametric excitation of a linear oscillator.Parametric amplifier186186186189191192199Chapter 13 Waves in semibounded media excited by perturbationsapplied to their boundaries20213.1 One-dimensional waves in non-linear homogeneous non-dispersivemedia. Shock and saw-tooth waves20213.2 One-dimensional waves in non-linear homogeneous slightly dispersivemedia described by the Korteweg-de Vries equation20613.3 One-dimensional waves in non-linear highly dispersive media20613.4 Non-linear wave bundles in dispersive media21113.4.1 Self-focusing and self-defocusing of wave bundles21113.4.2 Compression and expantion of pulses in non-linear dispersivemedia21613.5 Non-linear wave bundles in non-dispersive media. Approximatesolutions of the Khokhlov-Zabolotskaya equation21813.6 Waves in slightly inhomogeneous media220
13.7 Waves in periodically inhomogeneous mediaPart V OSCILLATIONS AND WAVES INACTIVE SYSTEMS. SELF-OSCILLATIONS ANDAUTO-WAVES223225Chapter 14 Forced oscillations and waves in active non-self-oscillatorysystems. Turbulence. Burst instability. Excitation of waves withnegative energy22714.1 Amplifiers with lumped parameters22714.2 Continuous semibounded media with convective instability22814.3 Excitation of turbulence in non-closed fluid flows. The Klimontovichcriterion of motion ordering22914.4 One-dimensional waves in active non-linear media. Burst instability . 23214.5 Waves with negative energy and instability caused by them235Chapter 15 Mechanisms of excitation and amplitude limitation ofself-oscillations and auto-waves. Classification of self-oscillatorysystems23915.1 Mechanisms of excitation and amplitude limitation of self-oscillationsin the simplest systems. Soft and hard excitation of self-oscillations . 23915.2 Mechanisms of the excitation of self-oscillations in systems with highfrequency power sources24115.3 Mechanisms of excitation of self-oscillations in continuous systems.Absolute instability as a mechanism of excitation of auto-waves . . . 24215.4 Quasi-harmonic and relaxation self-oscillatory systems. Stochasticand chaotic systems24215.5 Possible routes for loss of stability of regular motions and theappearance of chaos and stochasticity24315.5.1 The Feigenbaum scenario24315.5.2 The transition to chaos via fusion of a stable limit cycle withan unstable one and the subsequent disappearance of both ofthese cycles24415.5.3 The transition to chaos via destruction of a two-dimensionaltorus24415.5.4 The Ruelle-Takens scenario245C h a p t e r 16 E x a m p l e s of s e l f - o s c i l l a t o r y s y s t e m s w i t h l u m p e dparameters. I24616.1 Electronic generator. T h e van der Pol and Rayleigh equations . . . . 24616.2 T h e Kaidanovsky-Khaikin frictional generator and t h e Froudependulum25016.3 T h e Bonhoeffer-van der Pol oscillator25216.4 A model of glycolysis and a lumped version of t h e 'brusselator' . . . . 25316.5 A lumped model of t h e Buravtsev oscillator256
16.6 Clock movement mechanisms and the Neimark pendulum. Theenergetic criterion of self-oscillation chaotization16.7 Self-oscillatory models for species interaction based on theLotka-Volterra equations16.8 Systems with inertial non-linearity16.8.1 The Pikovsky model16.9 Systems with inertial excitation16.9.1 The Helmholtz resonator with non-uniformly heated walls16.9.2 A heated wire with a weight at its centre16.9.3 A modified 'brusselator'16.9.4 Self-oscillations of an air cushioned body259263264267267. . 270272276277Chapter 17 Examples of self-oscillatory systems with lumpedparameters. II28317.1 The Rossler and Chua systems28317.2 A three-dimensional model of an immune reaction illustrating an-oscillatory course of some chronic diseases. The 'oregonator' model . 28417.3 The simplest model of the economic progress of human society . . . . 28817.4 Models of the vocal source29317.5 A lumped model of the 'singing'flame303Chapter 18 Examples of self-oscillatory systems with high frequencypower sources30718.1 The Duboshinsky pendulum, a 'gravitational machine', and theAndreev hammer30718.2 The Bethenod pendulum, the Papaleksi effect, and the Rytov device . 31318.3 Electro-mechanical vibrators. Capacitance sensors of smalldisplacements317Chapter 19 Examples of self-oscillatory systems with time delay19.1 Biological controlled systems19.1.1 Models of respiration control19.1.2 The Mackey-Glass model of the process of regeneration ofwhite blood corpuscles (neutrophils)19.1.3 Models of the control of upright human posture19.2 The van der Pol-Duffing generator with additional delayed feedbackas a model of Doppler's autodyne19.3 A ring optical cavity with an external field (the Ikeda system) . . . .322322323329333336339Chapter 20 Examples of continuous self-oscillatory systems withlumped active elements34120.1 The Vitt system. Competition and synchronization of modes34120.2 The Rijke phenomenon34820.3 A distributed model of the 'singing'flame351
XIChapter 21 Examples of self-oscillatory systems with distributedactive elements35421.1 Lasers. Competition, synchronization and chaotization of modes.Optical auto-solitons35421.2 The Gann generators36821.3 Ionization waves (striations) in low temperature plasmas37421.3.1 Inert gases37821.3.2 Molecular gases38121.4 A model of the generation of. Korotkov's sounds38421.5 Self-oscillations of a bounded membrane resulting from excitation ofwaves with negative energy393Chapter 22 Periodic actions on self-oscillatory systems.Synchronization and chaotization of self-oscillations22.1 Synchronization of periodic self-oscillations by an external force in thevan der Pol-Duffing generator. Two mechanisms of synchronization.Synchronization as a non-equilibrium phase transition22.2 Synchronization of periodic oscillations in a generator with inertialnon-linearity and in more complicated systems22.3 Synchronization of a van der Pol generator with a modulated naturalfrequency22.4 Asynchronous quenching and asynchronous excitation of periodicself-oscillations22.5 Chaotization of periodic self-oscillations by a periodic external force22.6 Synchronization of chaotic self-oscillations. The synchronizationthreshold and its relation to the quantitative characteristics of theattractorChapter 23 Interaction between self-oscillatory systems23.1 Mutual synchronization of two generators of periodic oscillations . .23.2 Mutual synchronization of three and more coupled generators ofperiodic oscillations23.3 Chaotization of self-oscillations in systems of coupled generators . .23.4 Interaction between generators of periodic and chaotic oscillations .23.5 Interaction between generators of chaotic oscillations. The notion ofsynchronizationChapter 24 Examples of auto-waves and dissipative structures24.1 Auto-waves of burning. A model of a kink wave24.2 Auto-waves in the Fitz Hugh-Nagumo model24.3 Auto-waves in a distributed version of the brusselator and in someother models of biological, chemical and ecological systems24.4 Auto-waves described by the Kuramoto-Sivashinsky equation and thegeneralized Kuramoto-Sivashinsky equation396396401404409. 410412414.414421. 423. 424426431431434436440
XllChapter 25 Convective structures and self-oscillations in fluid. Theonset of turbulence44425.1 Rayleigh-Taylor instability and the initial stage of the excitation ofthermo-convection in a plane layer44425.2 Thermo-convection in a toroidal tube. The Lorenz equations45125.3 The initial stage of excitation of bio-convection45325.4 Onset of turbulence in theflowbetween two coaxial rotating cylinders.Taylor vortices456Chapter 26 Hydrodynamic and acoustic waves in subsonic jet andseparatedflows46326.1 The Kelvin-Helmholtz instability46326.2 Subsonic free jets46526.3 Sound excitation by an impinging jet. Excitation of edgetones . . . . 47726.4 Self-oscillations in open jet return circuit wind tunnels48126.5 The von Karman vortex wake, Aeolian tones and stalling flutter . . . 486Appendix A Approximate methods for solving linear differentialequations with slowly varying parameters489A.I JWKB Method489A.2 Asymptotic method490A.3 The Liouville-Green transformation491A.4 The Langer transformation492Appendix B The Whitham method and the stability of periodicrunning waves for the Klein—Gordon equation494Bibliography499Index535
Chapter 7 Natural oscillations of non-linear oscillators 71 7.1 Pendulum oscillations 71 7.2 Oscillations described by the Duffing equation 72 7.3 Oscillations of a material point in a force field with the Toda potential 75 7.4 Oscillations of a bubble in fluid 77 7.5 Oscillations of species strength described by the Lotka-Volterra equations 81
Introduction 1.1 Background in Nonlinear Oscillations Many phenomena associated with nonlinear oscillations, such as synchronizations, bifurcation phenomena, almost periodic oscillations, and chaotic oscillations, occur in nonlinear systems. In order to analyze the phenomena, we model the systems that exhibit the oscillations by nonlin-ear .
electromagnetic waves, like radio waves, microwaves, light, and x-rays are examples of transverse waves. Longitudinal waves travel through a medium in a direction parallel to the direction of travel of the wave. Mechanical waves such as sound waves, seismic waves created by earthquakes, and explosions are all examples of longitudinal waves.
Q: What are mechanical waves? A: Waves that require a medium in which to travel. A medium is the _ that waves travel through o Mediums can be solid, liquid, or gas Examples of mechanical waves include sound waves, seismic waves, ocean waves, etc Q: Describe two types of mechanical waves.
The nonlinear oscillations manifest themselves in various ways, depending on the initial conditions, and have a rich phenomenology. The study of neutrinos from these astrophysical sources therefore demands careful consideration of these nonlinear e ects. In this thesis, we put forward a framework to study nonlinear avor oscillations of neutrinos.
onlinear Rabi oscillations. Time-domain numerical simulations are run to observe both regimes. An analytical model is proposed to describe each regime and the transition from nonlinear to linear oscillations. This work confirms that the strong coupling between a single emitter and a
oscillations of nonlinear states have yet to be introduced. We report the Rabi oscillations of azimuthons— . 1 Introduction Rabi oscillations were introduced in quantum mechanics,1 but by now are widely investigated in a variety of optical and pho-tonic systems that include fibers,2,3 multimode waveguides,4-6
Nonlinear oscillations of viscoelastic microcantilever beam based on modi ed strain gradient theory . nonlinear curvature e ect, and nonlinear inertia terms are also taken into account. In the present study, the generalized derived formulation allows modeling any nonlinear . Introduction Microstructures have considerably drawn researchers' .
American Gear Manufacturers Association franklin@agma.org June 15, 2012. at Happened in the 2011 US Gear Market? mand for gears was up sharply in the US because of the mendous investment in “traditional” capital equipment. en though gear demand was up 28%, domestic shipments rose only %. The gap was filled by record gear imports (in terms of levels rowth), a 33% rise. ports were due to a .
