Chaotic Pendulum, The

By: Moshe Gitterman

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Hardcover | 21 September 2010

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158 Pages

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Pendulum is the simplest nonlinear system, which, however, provides the means for the description of different phenomena in Nature that occur in physics, chemistry, biology, medicine, communications, economics and sociology. The chaotic behavior of pendulum is usually associated with the random force acting on a pendulum (Brownian motion). Another type of chaotic motion (deterministic chaos) occurs in nonlinear systems with only few degrees of freedom. This book presents a comprehensive description of these phenomena going on in underdamped and overdamped pendula subject to additive and multiplicative periodic and random forces. No preliminary knowledge, such as complex mathematical or numerical methods, is required from a reader other than undergraduate courses in mathematical physics. A wide group of researchers, along with students and teachers will, thus, benefit from this definitive book on nonlinear dynamics.

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Non-Fiction Mathematics Applied Mathematics Chaos Theory Science Physics Classical Mathematics Gravity Statistical Physics Booktopia Publisher Services World Scientific Publishing

Preface	p. v
List of Equations	p. xi
Pendulum Equations	p. 1
Mathematical pendulum	p. 1
Period of oscillations	p. 5
Underdamped pendulum	p. 10
Nonlinear vs linear equation	p. 15
Isomorphic models	p. 16
Brownian motion in a periodic potential	p. 16
Josephson junction	p. 16
Fluxon motion in superconductors	p. 17
Charge density waves	p. 17
Laser gyroscope	p. 18
Synchronization phenomena	p. 18
Parametric resonance in anisotropic systems	p. 18
Phase-locked loop	p. 19
Dynamics of adatom subject to a time-periodic force	p. 19
The Frenkel-Kontorova model (FK)	p. 19
Solitons in optical lattices	p. 20
Other applications	p. 20
General concepts	p. 20
Phase space	p. 21
Poincare sections and strange attractors	p. 21
Lyapunov exponent	p. 22
Correlation function	p. 22
Spectral analysis	p. 22
Period doubling and intermittency	p. 23
Deterministic Chaos	p. 27
Damped, periodically driven pendulum	p. 27
Transition to chaos	p. 27
Two external periodic fields	p. 32
Dependence on driving frequency	p. 34
Role of damping	p. 35
Symmetry and chaos	p. 36
Diffusion in a chaotic pendulum	p. 39
Analytic methods	p. 41
Period-doubling bifurcations	p. 42
Melnikov method	p. 45
Parametric periodic force	p. 48
Pendulum with vertically oscillating suspension point	p. 49
Transition to chaos	p. 49
Melnikov method	p. 51
Parametric periodic non-harmonic force	p. 52
Downward and upward equilibrium configurations	p. 55
Boundary between locked and running solutions	p. 56
Pendulum with horizontally oscillating suspension point	p. 58
Pendulum with both vertical and horizontal oscillations of the suspension point	p. 62
Parametrically driven pendulum	p. 62
Periodic and constant forces	p. 65
Melnikov method	p. 66
Parametric and constant forces	p. 69
Harmonic balance method	p. 70
Heteroclinic and homoclinic trajectories	p. 71
Numerical calculations	p. 72
External and parametric periodic forces	p. 73
Pendulum subject to a Random Force	p. 77
Noise	p. 77
White noise and colored noise	p. 77
Dichotomous noise	p. 78
Langevin and Fokker-Planck equations	p. 79
External random force	p. 81
Constant and random forces	p. 82
External periodic and random forces	p. 85
Two sources of noise	p. 85
Fokker-Planck equation	p. 86
Ratchets	p. 86
Absolute negative mobility	p. 88
Pendulum with multiplicative noise	p. 89
Parametric periodic and random forces	p. 91
Damped pendulum subject to a constant torque, periodic force and noise	p. 92
Overdamped pendulum	p. 93
Additive white noise	p. 94
Additive dichotomous noise	p. 96
Multiplicative dichotomous noise	p. 99
Additive and multiplicative white noise	p. 102
Multiplicative dichotomous noise and additive white noise	p. 109
Correlated additive noise and multiplicative noise	p. 110
Systems with Two Degrees of Freedom	p. 113
Spring pendulum	p. 113
Dynamic equations	p. 114
Chaotic behavior of a spring pendulum	p. 118
Driven spring pendulum	p. 120
Double pendulum	p. 123
Spherical pendulum	p. 126
Conclusions	p. 131
Bibliography	p. 133
Glossary	p. 139
Index	p. 141
Table of Contents provided by Ingram. All Rights Reserved.

Chaotic Pendulum, The

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