Preface
Introduction
Chapter 1 Perturbed Oscillations
1.1 Linear Oscillator with Damping
1.2 Oscillator with Cubic Nonlinearity
1.3 Oscillator Under the Action of External Force. Resonance
1.4 A Forced Nonlinear Oscillator
1.5 Oscillators with Variable Parameters. Parametric Resonance
1.5.1 Slowly varying parameters. WKB approximation
1.5.2 Parametric resonance
1.6 Active Systems. The van der Pol Oscillator
1.7 A Lumped Model of Laser
1.8 Strongly Nonlinear Oscillators. A Pendulum
1.8.1 Ideal pendulum
1.8.2 Damping oscillations
1.9 A Charged Particle in the Magnetic Field
1.10 Interaction of Nonlinear Oscillators
1.11 Synchronization
1.11.1 Coupled Duffing oscillators
1.11.2 Synchronization of active oscillators
1.12 Self-Synchronization in Ensembles of Oscillators
1.12.1 Synchronization of limit cycles. Kuramoto model
1.12.2 Auto-synchronization of Duffing oscillators
1.13 Variable-Parameter Chaotic Oscillations
Appendix A. The Jacobi Elliptic Functions
Appendix B. Phase Plane
References
Chapter 2 Linear Waves
2.1 Kinematics of Waves. Phase and Group Velocity
2.2 Klein-Gordon Equation with Dissipation
2.2.1 Non-dissipative KG equation
2.2.2 KG with dissipation
2.3 Linear SchrSdinger Equation
2.3.1 General form
2.3.2 Gaussian impulse
2.4 Evolution of Wave Amplitude and Wavenumber
2.4.1 General equations
2.4.2 Self-similar solutions
2.4.3 Fresnel integrals
2.5 Asymptotic Behavior of Linear Waves
2.5.1 Method of stationary phase
2.5.2 Airy function
2.6 Wave Beams
2.6.1 Monochromatic beams
2.6.2 Space-time beams
2.7 Frequency-Modulated Dispersive Waves: Compression and Spreading
2.7.1 Space-time rays
2.7.2 Variation of wave energy and amplitude
2.7.3 Asymptotic of the envelope waves
2.8 Example: Water Waves
