## Chaotic Dynamics: An IntroductionInterest in chaotic dynamics has grown explosively in recent years. Applications to practically every scientific field have had far-reaching impact. As in the first edition, the authors present all the main features of chaotic dynamics using the damped, driven pendulum as the primary model. A special feature is the inclusion of both analytic and computer exercises with which the reader may expand upon the many numerical simulations included in the book. This allows learning through participation, without the extensive scientific background demanded by more advanced books. This second edition includes additional material on the analysis and characterization of chaotic data, and applications of chaos. Experimental data from a chaotic pendulum are analyzed using methods of nonlinear time series analysis. With the help of new computer programs provided in the book (and also available from one of the authors on an optional diskette), readers and students can learn about these methods and use them to characterize their own data. The second edition also explains methods for short-term prediction and control. Spatio-temporal chaos is now introduced with examples from fluid dynamics, crystal growth, and other areas. The number of references has more than doubled; solutions are included to selected exercises. This new edition of Chaotic dynamics can be used as a text for a unit on chaos for physics and engineering students at the second- and third-year level. Such a unit would fit very well into modern physics and classical mechanics courses. |

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### Contents

CHAPTER ONE Introduction | 8 |

CHAPTER THREE Visualization of the pendulums dynamics | 39 |

CHAPTER FOUR Toward an understanding of chaos | 74 |

CHAPTER FIVE The characterization of chaotic attractors | 109 |

CHAPTER SIX Experimental characterization prediction | 133 |

CHAPTER SEVEN Chaos broadly applied | 166 |

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### Common terms and phrases

algorithm angular velocity basins of attraction bifurcation diagram calculation CALL ADDLEGEND CALL CALL CALL CALL DATAGRAPH CALL SETXSCALE Cantor set chaotic attractor chaotic behavior chaotic dynamics chaotic pendulum chaotic systems Chapter circle map complex coordinates correlation dimension corresponding damping differential equations dissipative divergence drive cycles driven pendulum dynamical systems embedding dimension End IF LET end sub entropy example experimental data FINNUM fixed point fluid Fourier fractal frequency function graph GRAPHPoint illustrated infinite number initial conditions INITNUM INPUT PROMPT INPUT PROMPT"INPUT interval iteration Kolmogorov entropy laser LET xreal linear logistic map Lyapunov exponents method nonlinear dynamics number of points orbits oscillation oſt parameters period doubling phase diagram phase plane phase space PLOT Poincaré section power spectrum prediction PRINT reconstructed attractor region saddle point shown in Figure shows simulation spatio-temporal chaos tstep two-dimensional unstable values vector winding number ximag XINT xnew