## A Survey of Nonlinear Dynamics: "chaos Theory"This book is intended to give a survey of the whole field of nonlinear dynamics (or “chaos theory”) in compressed form. It covers quite a range of topics besides the standard ones, for example, pde dynamics and Galerkin approximations, critical phenomena and renormalization group approach to critical exponents. The many meanings or measures of “chaos” in the literature are summarized. A precise definition of chaos based on a carefully limited sensitive dependence is offered. An application to quantum chaos is made. The treatment does not emphasize mathematical rigor but insists that the crucial concepts and theorems be mathematically well-defined. Thus topology plays a basic role. This alone makes this book unique among short surveys, where the inquisitive reader must usually be satisfied with colorful similes, analogies, and hand-waving arguments.Richard Ingraham graduated with B.S. summa cum laude in mathematics from Harvard college and with M.A. and Ph.D in Physics from Harvard Graduate School. He was granted the Sheldon Prize Traveling Fellowship by Harvard College and was a member of the Institute for Advanced Study at Princeton for two years. |

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### Contents

PREFACE v | 1 |

ITERATION OF MAPS | 21 |

Vlll | 23 |

HAMILTONIAN SYSTEMS | 39 |

MEASURES OF CHAOS | 55 |

RENORMALIZATION GROUP | 65 |

PARTIAL DIFFERENTIAL EQUATIONS | 79 |

EXPERIMENTAL REALIZATIONS OF NONLINEAR DYNAMICS | 89 |

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### Common terms and phrases

algorithmic complexity analytic approximate asymptotically stable attractors autonomous flow behavior bifurcation diagram bifurcation value called cascade chaos chaotic const convection converge defined definition Df(x diagonal eigenvalue eigenvectors elliptic point example exists finite frequencies Galerkin method Hamiltonian system homeomorphism homoclinic hyperbolic fixed point ID map infinite intersect invariant Ising lattice iteration Jacobian matrix libration linear flow linear stability analysis logistic map Lyapunov exponent Lyapunov stable map dynamics map F minimal SD motion n-cycle neighborhood Vi C V nonintegrable nonlinear dynamics nonlinear flow normal modes orbit based orbit function parameter period-doubling bifurcation periodic orbit perturbed phase portrait phase space phase transition Poincare map power spectrum Prob Rayleigh number regime resonance zones saddle point sensitive dependence separatrices solution spins stability type stable 2-cycle stable fixed points stable subspace theorem topologically equivalent tori transverse unstable manifold vector Wiggins 36 zero

### References to this book

Approaches to the Qualitative Theory of Ordinary Differential Equations ... Tong-Ren Ding Limited preview - 2007 |