Oscillations and Waves: in Linear and Nonlinear Systems

One. Oscillations and Waves in Linear Systems.- 1. Linear Oscillators.- 1.1. General Notes.- 1.2. Two Examples. The Phase Plane Diagram of an Oscillator.- 1.3. Resonance. The Effect of an Aperiodic External Force on an Oscillator.- 1.4. Normal Oscillations. Analogy with Quantum Mechanics. Production and Extinction Operators.- 2. Oscillations in a System with Two Linked Oscillators.- 2.1. Initial Equations.- 2.2. The Fundamental Oscillations of Two Linked Oscillators.- 2.3. Disturbance of Two Linked Oscillators by an External Force. The Reciprocity Principle.- 3. Oscillations in an Ensemble of Non-Interacting Oscillators.- 3.1. Classical Theory of Dispersion.- 3.2. Oscillations in an Ensemble of Dissimilar Noninteracting Oscillators with a Given Distribution Function.- 4. Oscillations in Ordered Structures. Limit for a Continuous medium. Waves. Dispersion.- 4.1. General Remarks.- 4.2. Oscillations in Ordered Structures (Chains of Linked Particles and Identical Linked Oscillators).- 4.3. Limiting Transition from an Ordered Structure to a One-dimensional Medium. Temporal and Spatial Dispersion. Physical Nature of Dispersion.- 4.4. Typical Dispersion Characteristics for Medium Models.- 4.5. Formal Method for Obtaining the Dispersion Equation. Waves in a One-Dimensional Resonator. Resonance in Wave Systems.- 5. Properties of Waves with Small Amplitudes in Continuous media.- 5.1. General Remarks.- 5.2. Equations of Hydrodynamics. Dispersion for Sound Waves. For Sound Waves.- 5.3. A Stratified Fluid. Sound in an Ocean.- 5.4. Gravity Waves in an Incompressible Liquid. Internal Waves. Rossby Waves.- 5.5. Waves in a Superfluid Liquid.- 5.6. Waves in a Plasma. Hydrodynamic Description.- 6. Stability and Instability of Linear Systems with Discrete Spectra.- 6.1. General Notes and Definitions.- 6.2. The Raus-Gurvits Criterion and Three-Dimensional Systems.- 6.3. The D-Partition Method.- 6.4. Stability of Non-Autonomous Systems.- 6.5. Instability Mechanisms.- 7. Stability of Distributed Systems with Continuous Spectra.- 7.1. General Comments.- 7.2. Examples of Instability.- 7.3. Absolute and Convective Instability. The Characteristics Method.- 7.4. Waves in Flows. Electron Beams. Helmholtz Instability.- 7.5. Amplification and Filtering. Separation Criteria.- 8. Propagation Velocity of Waves.- 8.1. Various Introductions to the Concept of Group Velocity.- 8.2. Group Velocity of Waves in Some Continuous Media.- 9. Energy and Momentum of Waves.- 9.1. Equation for the Transport of the Average Energy Density by Wave Packets in Dispersing Media.- 9.2. Density of the Energy of an Electromagnetic Wave in a Medium with Dispersion.- 9.3. Momentum of a Wave Packet.- 10. Waves with Negative Energy. Linked Waves.- 10.1. General Notes.- 10.2. Waves with Positive and Negative Energies.- 10.3. Coupled Waves. Synchronicity. Normal and Anomalous Doppler Effects.- 11. Parametric Systems and Parametric Instability.- 11.1. General Comments.- 11.2. Parametric Resonance. Floquet's (Blokh's) Theorem. Mathieu's Equation.- 11.3. Waves in Periodic Structures. The Mathieu Zone and the Brillouin Diagram.- 11.4. Motion in a Rapidly Oscillating Field. Kapitsa's Pendulum. Free Electron Lasers.- 12. Adiabatic Invariants. Propagation of Waves in Inhomogeneous Media.- 12.1. The Wentsel-Kramers-Brillouin (VCB) Approximation and Adiabatic Invariants.- 12.2. Equivalence Between a Rotor and an Oscillator.- 12.3. Propagation of Waves in Inhomogeneous Media. The Approximation of Geometric Optics.- 12.4. The Propagation of Waves in a Plane-Layer Medium in the Geometric Optics Approximation.- 12.5. Linear Wave Interaction in an Inhomogeneous Medium.- Two. Oscillations and Waves in Nonlinear Systems.- 13. The Nonlinear Oscillator.- 13.1. Initial remarks.- 13.2. Qualitative and Analytical Description. Examples of Nonlinear Systems.- 13.3. Nonlinear Resonance.- 13.4. Overlap between Nonlinear Resonances.- 14. Periodic Self-Excited Oscillations.- 14.1. Definitions.- 14.2. The Van der Pol Generator. Self-Excited Oscillations as a Function of System Parameters.- 14.3. Relaxational Self-Excited Oscillations. Fast and Slow Motions.- 15. General Properties of Nonlinear Dynamic Systems in Phase Space.- 15.1. Basic Types of Trajectory. The Fundamentals of Dynamic Systems (Structural Stability).- 15.2. Basic Bifurcations on a Plane. Poincare Indices.- 15.3. Point Transformations.- 15.4. Bifurcation of Periodic Motions.- 15.5. Homoclinic Structures.- 16. Self-Excited Oscillations in Multifrequency Systems.- 16.1. Forced Synchronization.- 16.2. Competition.- 16.3. Mutual Mode Synchronization.- 17. Resonance Interactions between Oscillators.- 17.1. Interaction Between Three Coupled Oscillators in a System with Quadratic Nonlinearity.- 17.2. Resonance Interactions Between Waves in Weakly Nonlinear Media with Dispersion.- 17.3. Explosive Instability.- 18. Simple Waves and the Formation of Discontinuities.- 18.1. Kinematic Waves.- 18.2. Travelling Waves in a Nonlinear Medium Without Dispersion.- 18.3. Determining the Discontinuity Coordinates.- 18.4. Weak Shock Waves. Boundary Conditions at a Discontinuity.- 19. Stationary Shock Waves and Solitons.- 19.1. Structure of a Discontinuity.- 19.2. Solitary Waves - Solitons.- 19.3. Solitons as Particles.- 19.4. Higher-Dimensional Solitons.- 20. Modulated Waves in Nonlinear Media.- 20.1. General Remarks.- 20.2. Self-Modulation. Reversibility.- 20.3. Self-Focusing.- 20. 4. Interaction Between Wave Beams and Packets.- 20.5. Interactions Between Waves Having Randomly Modulated Phases. Wave Kinetics.- 21. Self-Excited Oscillations in Distributed Systems.- 21.1. General Remarks.- 21.2. Medium Without Dispersion. Discontinuous Waves.- 21.3. Stationary Waves.- 21.4. The Existence and Role of Limiting Cycles.- 21.5. Competition Between Stationary Waves in an Active Medium.- 21.6. Periodic Self-Excited Oscillations in Hydrodynamic Flows.- 22. Stochastic Dynamics in Simple Systems.- 22.1. How Randomness Appears in a Dynamic System.- 22.2. The Stochastic Dynamics of One-Dimensional Mappings.- 22.3. Noise Generator. Qualitative Description and Experiment.- 22.4. Statistical Description of a Simple Noise Generator.- 22.5. Ways in which Strange Attractors Arise.- 22.6. Dimensionality of Stochastic Sets.- 23. The Onset of Turbulence.- 23.1. General Remarks.- 23.2. The Occurrence of Stochastic Self-Excited Oscillations in Experimental Fluid Mechanics.- 23.3. Stochastic Modulation.- 23.4. Ideal Flow and Turbulence.- 24. Self-Organization.- 24.1. Main Phenomena, Models, and Mathematical Forms.- 24.2. Travelling Pulsations.- 24.3. Spiral and Cylindrical Waves. Travelling Centers.- 24.4. Concerning Self-Organization Mechanisms.- References.