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Oscillations and Waves :  In Linear and Nonlinear Systems - M. I. Rabinovich

Oscillations and Waves

In Linear and Nonlinear Systems

Hardcover Published: 15th November 1989
ISBN: 9780792304456
Number Of Pages: 578

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'Et mai *...* si j'avait su comment en revenir. One service mathematics has rendered the je n'y semis point aUe.' human race. It has put common sense back Jules Verne where it belongs, on the topmost sheJf next to the dusty canister Iabclled 'discarded non* The series is divergent; therefore we may be sense'. Eric T. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

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.

ISBN: 9780792304456
ISBN-10: 0792304454
Series: Mathematics and Its Applications
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 578
Published: 15th November 1989
Country of Publication: NL
Dimensions (cm): 23.39 x 15.6  x 3.33
Weight (kg): 1.01

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