## The Chaotic PendulumPendulum 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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### Contents

Chapter 2 Deterministic Chaos | 27 |

Chapter 3 Pendulum subject to a Random Force | 77 |

Chapter 4 Systems with Two Degrees of Freedom | 113 |

Chapter 5 Conclusions | 131 |

133 | |

Glossary | 139 |

141 | |

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### Common terms and phrases

additive noise analytical angle average angular velocity bias Bifurcation diagram chaotic behavior chaotic motion chaotic solutions control parameter Copyright 2006 deﬁned described deterministic chaos dichotomous noise different values differential equations diffusion downward position driven pendulum driving force driving frequency dynamics elliptic equation of motion equilibrium external ﬁeld external force external periodic ﬁnd ﬁrst ﬁxed point ﬂuctuations Fokker-Planck equation harmonic homoclinic orbit increases initial conditions Josephson junction Langevin equation linear Lyapunov exponent Melnikov function Melnikov method multiplicative noise nonlinear numerical calculations numerical solution obtained onset of chaos overdamped pendulum equation pendulum subject period-doubling bifurcations periodic potential perturbation phase space plane Poincare section power spectrum random force ratchet regime region Reprinted with permission resonance rotations running solutions separatrix shown in Fig sinQ5 solution of Eq spring pendulum stable strange attractors suspension point symmetric torque transition to chaos vertical oscillations whereas white noise zero