## Chaos: An Introduction to Dynamical SystemsBACKGROUND Sir Isaac Newton hrought to the world the idea of modeling the motion of physical systems with equations. It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, of position. His greatest single success was his discovery that which are derivatives the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the hodies. He demonstrated that the ohserved motion of the planets could he explained hy assuming that there is a gravitational attraction he tween any two ohjects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them. The circular, elliptical, and parabolic orhits of astronomy were v INTRODUCTION no longer fundamental determinants of motion, but were approximations of laws specified with differential equations. His methods are now used in modeling motion and change in all areas of science. Subsequent generations of scientists extended the method of using differ ential equations to describe how physical systems evolve. But the method had a limitation. While the differential equations were sufficient to determine the behavior-in the sense that solutions of the equations did exist-it was frequently difficult to figure out what that behavior would be. It was often impossible to write down solutions in relatively simple algebraic expressions using a finite number of terms. Series solutions involving infinite sums often would not converge beyond some finite time. |

### Contents

1 | |

17 | |

COUNTING THE PERIODIC ORBITS | 92 |

CHAOS | 108 |

FRACTALS | 149 |

CHAOS IN TWODIMENSIONAL MAPS | 194 |

5 | 212 |

CHAOTIC ATTRACTORS | 231 |

PERIODIC ORBITS AND LIMIT SETS | 329 |

CHAOS IN DIFFERENTIAL EQUATIONS | 359 |

STABLE MANIFOLDS AND CRISES | 399 |

BIFURCATIONS | 447 |

CASCADES | 499 |

STATE RECONSTRUCTION FROM DATA | 537 |

Adaptive StepSize Methods | 574 |

BIBLIOGRAPHY | 587 |

### Other editions - View all

Chaos: An Introduction to Dynamical Systems Kathleen T. Alligood,Tim D. Sauer,James A. Yorke Limited preview - 2006 |

Chaos: An Introduction to Dynamical Systems Kathleen T. Alligood,Tim D. Sauer,James A. Yorke Limited preview - 2000 |

Chaos: An Introduction to Dynamical Systems Kathleen T. Alligood,Tim D. Sauer,James A. Yorke No preview available - 2000 |

### Common terms and phrases

A₁ attracting AXIS basin behavior bifurcation diagram boundary box-counting dimension boxes Cantor set cat map chaotic attractor chaotic orbit Chapter circle contains converge coordinates corresponding countable cross curves defined definition denoted derivative differential equation disk dynamics eigenvalues eigenvector ellipse equilibrium example EXERCISE forward limit set fractal Hénon map infinite initial conditions initial value integer intersects invariant set iterates itinerary Jacobian laser Lemma Let f linear map logistic map Lorenz Lyapunov exponent Lyapunov function Lyapunov number map f map f(x matrix neighborhood nonlinear one-dimensional map origin period-doubling bifurcation period-k period-three period-two orbit periodic orbit periodic points phase plane plotted Poincaré Poincaré-Bendixson Theorem rectangle region S₁ saddle-node bifurcation sensitive dependence sequence shown in Figure shows sink solution stable and unstable subintervals tent map Theorem torus trajectory transition graph two-dimensional unit interval unit square unstable manifolds vector vertical w-limit set w(vo zero