Oscillations and waves

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IntroductionDownload the introductionВ.1. Subject of oscillations and waves theory
В.2. Classification of the dynamical systems
В.2.1. Classification after the number of the freedom degrees. В.2.2. Classification after the energetic characteristics. В.2.3. Classification after the character of differential equation. В.2.4. Conventional character of the real oscillating systems' classification.
Part I.Dynamical systems with one degree of freedom.
Section 1.1.Download part 1.1Free oscillations of the linear oscillator.
1.1.1. Examples of linear oscillators. 1.1.2. Solution of linear dissipative oscillator equation. 1.1.3. Equation of conservative oscillator.
Section 1.2.Download part 1.2Forced oscillations of the linear oscillator for harmonic force.
1.2.1. Examples of forced oscillations of the linear oscillators. 1.2.2. Oscillator under operation of the external force as a system with 3/2 degrees of freedom. 1.2.3. Oscillations of linear conservative oscillator under operation of the harmonic force. 1.2.4. Dissipation influence on the forced harmonic oscillations.
Section 1.3. Download part 1.3Forced oscillations of linear oscillator under operation of arbitrary force: methods of analysis.
1.3.1. Superposition principle. 1.3.2. Method of Fourier series and Fourier integrals. 1.3.3. Integral of Duhamel. 1.3.4. Interrelation between transfer function and Green function. 1.3.5. Green function's presentation trough eigen functions of homogeneous equations.
Section 1.4.Download part 1.4Elementary theory of parametrical resonance (for stepped pumping).
1.4.1. Parametric resonance in the conservative system. 1.4.2. Parametric resonance in the dissipative system. 1.4.3. Dissipative parametric oscillator as a regenerative amplifier. 1.4.4. Conditions of validity of parametric approximation.
Section 1.5.Download part 1.5Parametric generator with harmonic pumping.
1.5.1. Obtaining of the Mathieu equation. 1.5.2. Approximate solution of the Mathieu equation using the slow amplitudes method. 1.5.3. Mechanism of parametric instability. 1.5.4. Zones of instability for the Mathieu equation.
Section 1.6.Download part 1.6Single-tuned parametric amplifier.
1.6.1. Solution of the heterogeneous Mathieu equation. 1.6.2. Frequency response function. 1.6.3. Amlpification coefficient's pulsing. 1.6.4. Dependence of amplification coefficient upon the signal phase.
Section 1.7.Download part 1.7Free oscillations of the non-linear oscillator.
1.7.1. Conservative oscillator with quadratic and cubic non-linearity: analysis by the method of prograssive approximations. 1.7.2. Anharmonicity and anisochronism of the non-linear oscillator. 1.7.3. Phase portrait of the non-linear oscillator.
Section 1.8.Download part 1.8Stability of oscillation system with the lumped parameters.
1.8.1. Stability of the motion after Lyapunov. 1.8.2. Orbital stability. 1.8.3. Asymptotic and absolute stability. 1.8.4. Criterion of Routh - Hurwits. 1.8.5. Structural stability (roughness) of dynamical system.
Section 1.9.Download part 1.9Forced oscillations of the non-linear dissipative oscillator.
1.9.1. Discussion of the model. 1.9.2. The used approximations and obtaining the equation for the amplitude. 1.9.3. Non-linear restriction of the resonance oscillations' amplitude. 1.9.4. Graphical analysis of the amplitude equation. 1.9.5. Hysteresis. 1.9.6. Influence of the weak dissipation on the hysteresis phenomenon. 1.9.7. Resonance curve for the dissipative oscillator. 1.9.8. Resonance on the divisible harmonics and on the subharmonics.
Section 1.10.Download part 1.10Forced oscillations of the non-linear conservative oscillator.
1.10.1. Hamilton systems and variables operation-angle. 1.10.2. Potential of the external force. 1.10.3. Equations for the non-linear resonance. 1.10.4. Phase oscillations. 1.10.5. Width of the non-linear resonance. 1.10.6. Analysis of the used approximations. 1.10.7. Overlap of the non-linear resonances. 1.10.8. Forced oscillations of the mathematical pendulum near the separatrix. 1.10.9. Stochastic layer. 1.10.10. Influence of the non-resonant terms on the phase oscillations.
Section 1.11.Download part 1.11Self-oscillations.
1.11.1. Obtaining of the non-linear equation for Van der Pol self-oscillator. 1.11.2. Conditions of the generator's auto-excitation. 1.11.3. Rayleigh equations: qualitative analysis of the solution. 1.11.4. Regime of quasi-harmonic oscillations. 1.11.5. Phase portrait of quazi-harmonic oscillations' generator. 1.11.6. Regime of the relaxation oscillations.
Section 1.12.Download part 1.12Oscillations of self-oscillator under operation of the external force.
1.12.1. Quasi-linear theory of the self-oscillator and effect of the forced sinchronization. 1.12.2. Van der Pol heterogeneous equation: solution by the method of the slow amplitudes. 1.12.3.Resonance curves for self-oscillator forced oscillations. 1.12.4. Stability of the resonance curves. 1.12.5. Forced sinchronization: case of the small external forces. 1.12.6. Forced sinchronization: case of the strong external forces. 1.12.7. Forced sinchronization by the half and double frequency.
Section 1.13.Download part 1.13Stochastic regime of the dynamical system: Kiyashko - Pikovsky - Rabinovich (KPR) noise generator.
1.13.1. Unexpected behaviour of the simple systems. 1.13.2. KPR noise generator: the circuit and the motion equation. 1.13.3. Qualitative analysis of the motion equations for the regular oscillations' regimes. 1.13.4. Phase portrait of the stochastic oscillations. 1.13.5. Bifurcations of the KPR generator under the change of driving parameter.
Part ІІ.Dynamical systems with many degrees of freedom.
Section 2.1.Download part 2.1 Free oscillations in the system with many degrees of freedom.
2.1.1. Examples of the systems with many degrees of freedom. 2.1.2. Equation of motion for coupled conservative oscillators. 2.1.3. Charactersic equation, eigen frequencies and coefficients of the amplitude distribution. 2.1.4. Energy exchange among the degrees of freedom. 2.1.5. Normal modes. 2.1.6. Eigen frequencies repulsion.
Section 2.2.Download part 2.2Forced oscillations in the linear systems with many degrees of freedom.
2.2.1. Basic equations. 2.2.2. Peculiarities of resonance curves. 2.2.3. Reciprocal theorem. 2.2.4. Orthogonality of the external forces to the normal modes.
Section 2.3.Download part 2.3Oscillations of the parametric systems with many degrees of freedom.
2.3.1. Two-circuit parametric amplifier: the basic equations. 2.3.2. Regenerative amplifier with the high-frequency pumping. 2.3.3. Non-regenerative up-converter with the low-frequency pumping. 2.3.4. Manley - Rowe relation.
Section 2.4.Download part 2.4Modes' concurrence in the multi-frequency self-oscillator.
2.4.1. The circuit of the two-frequency self-oscillator. 2.4.2. Basic equations. 2.4.3. Obtaining of the reduced equations. 2.4.4. Equations set for modes with the non-linear coupling. 2.4.5. Stationary solutions and their stability. 2.4.6. Phase portraits. 2.4.7. Frequency pulling in the two-frequency self-oscillator.
Section 2.5.Download part 2.5Chaos in the Hamilton systems with many degrees of freedom.
2.5.1. Operator of the stream. 2.5.2. Phase space for the Hamilton systems. 2.5.3. Ergodic systems. 2.5.4. Mixing in the Hamilton systems. 2.5.5. Lyapunov exponents and entropy of Kolmogorov - Sinay. 2.5.6. Spectra of the chaotic motion.
Section 2.6.Download part 2.6Chaos in the dissipative systems with many degrees of freedom.
2.6.1. Simple attractors. 2.6.2. Strange attractors. 2.6.3. Fractals. 2.6.4. Scenarios of the transition to chaos. 2.6.5. Scenario Ruelle - Takens. 2.6.6. Scenario Feigenbaum. 2.6.7. Scenario of Pomeau - Mannerville.
Part 2.7.Download part 2.7Chaos in the dissipative systems with many degrees of freedom.
Part 2.8.Download part 2.8Chain systems.
Part ІІІ.Systems with the distributed parameters.
Section 3.1.Download part 3.1Linear waves in the passive systems with the distributed parameters.
3.1.1. Examples of the small amplitude waves in the passive systems. 3.1.2. Solution of the wave equation. Dispersion. 3.1.3. Initial and boundary problem. 3.1.4. Propagation of the wave packet in the line with dispersion. 3.1.5. Formal classification of the dispersion. 3.1.6. Reasons of dispersion's existance (model treating). 3.1.7. Space and time dispersion in the Electrodynamics of the continuum media.
Section 3.2.Download part 3.2Bound modes in the passive systems.
3.2.1.Bound modes without dispersion. 3.2.2. Bound modes with the same dispersion signs. 3.2.3. Bound modes with the different dispersion signs.
Section 3.3.Download part 3.3Bound modes in the open (non-equilibrium) systems.
3.3.1. Dispersion equations for waves in the two beams amplifier. 3.3.2. Generator on the coming beams. 3.3.3. Two beams amplifier. 3.3.4. Absolute, convection and oscillator instability. 3.3.5. Briggs criterion.
Section 3.4.Download part 3.4Non-linear waves in the passive systems.
3.4.1. Waves in the conservative non-linear media without dispersion. 3.4.2. Waves in the non-linear weakly dissipated media. 3.4.3. Waves in the non-linear conservative media with the weak dispersion.
Section 3.5.Download part 3.5Waves in the non-linear active systems.
3.5.1. Diffusion type non-linear kinetics equations. 3.5.2. Bistable media. Running fronts. 3.5.3. Regenerating media. Running pulses and spiral waves. 3.5.4. Self-oscillating media. Phase waves. 3.5.5. Dissipative structures. 3.5.6. Turbulence.
Part 3.6.Download part 3.6Waves in non-linear passive systems with the weak dispersion.
Part 3.7.Download part 3.7Waves in the non-linear active systems.
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