## Chaos: An Introduction to Dynamical SystemsChaos: An Introduction to Dynamical Systems, was developed and class-tested by a distinguished team of authors at two universities through their teaching of courses based on the material. Intended for courses in nonlinear dynamics offered either in Mathematics or Physics, the text requires only calculus, differential equations, and linear algebra as prerequisites. Along with discussions of the major topics, including discrete dynamical systems, chaos, fractals, nonlinear differential equations and bifurcations, the text also includes Lab Visits, short reports that illustrate relevant concepts from the physical, chemical and biological sciences. There are Computer Experiments throughout the text that present opportunities to explore dynamics through computer simulations, designed to be used with any software package. And each chapter ends with a Challenge, which provides students a tour through an advanced topic in the form of an extended exercise. |

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### Contents

INTRODUCTION | 1 |

TWODIMENSIONAL MAPS | 43 |

CHAOS | 105 |

FRACTALS | 149 |

CHAOS IN TWODIMENSIONAL MAPS | 193 |

CHAOTIC ATTRACTORS | 231 |

DIFFERENTIAL EQUATIONS | 273 |

PERIODIC ORBITS AND LIMIT SETS | 329 |

BIFURCATIONS | 447 |

CASCADES | 499 |

STATE RECONSTRUCTION FROM DATA | 537 |

A MATRIX ALGEBRA | 557 |

B COMPUTER SOLUTION OF ODES | 567 |

ANSWERS AND HINTS TO SELECTED EXERCISES | 577 |

BIBLIOGRAPHY | 587 |

595 | |

### Common terms and phrases

approximately Assume attracting baker map basin behavior bifurcation diagram bifurcation orbit boundary box-counting dimension branch Cantor set cascade cat map chaotic attractor chaotic orbit Chapter circle contains converge coordinates corresponding crosses curves defined definition denoted derivative differential equation disk eigenvalues eigenvector ellipse equilibrium example EXERCISE finite fractal Henon map infinite initial conditions initial value integer intersect iterates itinerary Jacobian laser Lemma Let f linear map logistic map Lorenz Lyapunov exponent Lyapunov function Lyapunov number map f matrix measure neighborhood nonlinear one-dimensional maps one-to-one origin parameter value path pendulum period-doubling bifurcation period-k period-three period-two orbit periodic orbit periodic points phase plane plot Poincare rectangle saddle-node bifurcation sensitive dependence sequence shown in Figure shows sink solution stable and unstable Step subintervals subset tent map Theorem torus trajectory transition graph two-dimensional unit interval unit square unstable manifolds unstable orbits vector vertical w-limit set zero