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Chaotic Pendulum, The - Moshe Gitterman

Chaotic Pendulum, The


Published: 7th September 2010
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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.

Prefacep. v
List of Equationsp. xi
Pendulum Equationsp. 1
Mathematical pendulump. 1
Period of oscillationsp. 5
Underdamped pendulump. 10
Nonlinear vs linear equationp. 15
Isomorphic modelsp. 16
Brownian motion in a periodic potentialp. 16
Josephson junctionp. 16
Fluxon motion in superconductorsp. 17
Charge density wavesp. 17
Laser gyroscopep. 18
Synchronization phenomenap. 18
Parametric resonance in anisotropic systemsp. 18
Phase-locked loopp. 19
Dynamics of adatom subject to a time-periodic forcep. 19
The Frenkel-Kontorova model (FK)p. 19
Solitons in optical latticesp. 20
Other applicationsp. 20
General conceptsp. 20
Phase spacep. 21
Poincare sections and strange attractorsp. 21
Lyapunov exponentp. 22
Correlation functionp. 22
Spectral analysisp. 22
Period doubling and intermittencyp. 23
Deterministic Chaosp. 27
Damped, periodically driven pendulump. 27
Transition to chaosp. 27
Two external periodic fieldsp. 32
Dependence on driving frequencyp. 34
Role of dampingp. 35
Symmetry and chaosp. 36
Diffusion in a chaotic pendulump. 39
Analytic methodsp. 41
Period-doubling bifurcationsp. 42
Melnikov methodp. 45
Parametric periodic forcep. 48
Pendulum with vertically oscillating suspension pointp. 49
Transition to chaosp. 49
Melnikov methodp. 51
Parametric periodic non-harmonic forcep. 52
Downward and upward equilibrium configurationsp. 55
Boundary between locked and running solutionsp. 56
Pendulum with horizontally oscillating suspension pointp. 58
Pendulum with both vertical and horizontal oscillations of the suspension pointp. 62
Parametrically driven pendulump. 62
Periodic and constant forcesp. 65
Melnikov methodp. 66
Parametric and constant forcesp. 69
Harmonic balance methodp. 70
Heteroclinic and homoclinic trajectoriesp. 71
Numerical calculationsp. 72
External and parametric periodic forcesp. 73
Pendulum subject to a Random Forcep. 77
Noisep. 77
White noise and colored noisep. 77
Dichotomous noisep. 78
Langevin and Fokker-Planck equationsp. 79
External random forcep. 81
Constant and random forcesp. 82
External periodic and random forcesp. 85
Two sources of noisep. 85
Fokker-Planck equationp. 86
Ratchetsp. 86
Absolute negative mobilityp. 88
Pendulum with multiplicative noisep. 89
Parametric periodic and random forcesp. 91
Damped pendulum subject to a constant torque, periodic force and noisep. 92
Overdamped pendulump. 93
Additive white noisep. 94
Additive dichotomous noisep. 96
Multiplicative dichotomous noisep. 99
Additive and multiplicative white noisep. 102
Multiplicative dichotomous noise and additive white noisep. 109
Correlated additive noise and multiplicative noisep. 110
Systems with Two Degrees of Freedomp. 113
Spring pendulump. 113
Dynamic equationsp. 114
Chaotic behavior of a spring pendulump. 118
Driven spring pendulump. 120
Double pendulump. 123
Spherical pendulump. 126
Conclusionsp. 131
Bibliographyp. 133
Glossaryp. 139
Indexp. 141
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814322003
ISBN-10: 9814322008
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 156
Published: 7th September 2010
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 22.86 x 15.24  x 1.91
Weight (kg): 0.41