## The Chaos Avant-garde: Memories of the Early Days of Chaos Theory (Google eBook)This book is an authoritative and unique reference for the history of chaos theory, told by the pioneers themselves. It also provides an excellent historical introduction to the concepts. There are eleven contributions, and six of them are published here for the first time OCo two by Steve Smale, three by Yoshisuke Ueda, and one each by Ralph Abraham, Edward Lorenz, Christian Mira, Floris Takens, T Y Li and James A Yorke, and Otto E Rossler. Contents: On How I Got Started in Dynamical Systems 1959OCo1962 (S Smale); Finding a Horseshoe on the Beaches of Rio (S Smale); Strange Attractors and the Origin of Chaos (Y Ueda); My Encounter with Chaos (Y Ueda); Reflections on the Origin of the Broken-Egg Chaotic Attractor (Y Ueda); The Chaos Revolution: A Personal View (R Abraham); The Butterfly Effect (E Lorenz); I Gumowski and a Toulouse Research Group in the OC PrehistoricOCO Times of Chaotic Dynamics (C Mira); The Turbulence Paper of D Ruelle & F Takens (F Takens); Exploring Chaos on an Interval (T Y Li & J A Yorke); Chaos, Hyperchaos and the Double-Perspective (O E RAssler). Readership: Educators and university students of science and mathematics." |

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### Contents

7 | |

Strange Attractors and the Origin of Chaos | 23 |

My Encounter with Chaos | 57 |

Reflections on the Origin of the BrokenEgg Chaotic Attractor | 65 |

A Personal View | 81 |

The Butterfly Effect | 91 |

The Turbulence Paper of D Ruelle and F Takens | 199 |

Exploring Chaos on an Interval | 201 |

Sources | 221 |

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analog computer analog computer experiments Andronov Applications basin boundary Berkeley bifurcation Birkhoff broken-egg attractor chaos theory chaotic attractor chaotic dynamics circuit CNRS complex Comptes Rendus Acad corresponding David Ruelle deuxième ordre domain Duffing equation engineering Figure fractal giving rise Gumowski harmonic Hayashi homoclinic points horseshoe infinitely International invariant closed curve invariant curves island structure iterated Japanese Kawakami Kyoto University laboratory lectures Lefschetz linéaire linear Lorenz Math mathematicians mathematics method Mira motion Myrberg noninvertible maps nonlinear dynamics Nonlinear Oscillations obtained one-dimensional ordinary differential equations paper parameter values Paris Peixoto period doubling periodic solution perturbation phase phenomena phenomenon Poincare preimages Princeton problem Prof Professor published quadratic map qualitative Ralph Abraham récurrence Ruelle saddle fixed point Sciences simulation Smale stable and unstable stable fixed point Steve Smale stochastic structural stability studies Thom topology Toulouse group Transformations Ponctuelles two-dimensional Ueda unstable cycles unstable manifolds Yoshisuke Ueda

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Page 5 - ... the only go player in the Soviet Union. My roommate in Kiev was Larry Markus. I met and saw much of Anosov in Kiev. Anosov had followed the Gorki school, but he was based in Moscow. After Kiev I went back to Moscow where Anosov introduced me to Arnold, Novikov, and Sinai. I must say I was extraordinarily impressed to meet such a powerful group of four young mathematicians. In the following years, I often said there was nothing like that in the West. I gave some lectures at the Steklov Institute...

Page 3 - Morse inequalities for a class of dynamical systems incorporating it. However, my overenthusiasm led me to suggest in the paper that these systems were almost all (an open dense set) of ordinary differential equations! If I had been at all familiar with the literature (Poincaré, Birkhoff, Cartwright-Littlewood), I would have seen how crazy this idea was. On the other hand, these systems, though sharply limited, would find a place in the literature, and were christened Morse-Smale dynamical systems...