## An Introduction to Chaotic Dynamical SystemsThe study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry. Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas. |

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### Contents

1 | |

2 | |

8 | |

17 | |

24 | |

31 | |

16 Symbolic dynamics | 39 |

17 Topological conjugacy | 44 |

21 Preliminaries from linear algebra and advanced calculus | 161 |

23 The horseshoe map | 181 |

24 Hyperbolic toral automorphisms | 190 |

25 Attractors | 201 |

26 The stable and unstable manifold theorem | 214 |

27 Global results and hyperbolic sets | 232 |

28 The Hopf bifurcation | 240 |

29 The Hénon map | 251 |

18 Chaos | 49 |

19 Structural stability | 53 |

110 Sarkovkiis theorem | 60 |

111 The Schwarzian derivative | 69 |

112 Bifurcation theory | 80 |

113 Another view of period three | 93 |

114 Maps of the circle | 102 |

115 MorseSmale diffeomorphisms | 114 |

116 Homoclinic points and bifurcations | 122 |

117 The perioddoubling route to chaos | 130 |

118 The kneading theory | 139 |

119 Genealogy of periodic points | 147 |

Higher Dimensional Dynamics | 159 |

Complex Analytic Dynamics | 260 |

31 Preliminaries from complex analysis | 261 |

32 Quadratic maps revisited | 268 |

33 Normal families and exceptional points | 272 |

34 Periodic points | 276 |

35 The Julia set | 283 |

36 The geometry of Julia sets | 289 |

37 Neutral periodic points | 300 |

38 The Mandelbrot set | 311 |

the exponential function | 319 |

Color Plates | 329 |

Index | 332 |

### Common terms and phrases

analytic maps assume attracting fixed point attractor basin of attraction behavior bifurcation diagram Cantor set chain recurrent chapter converges critical point curve define Definition denote dense depicted in Fig diffeomorphism disk dynamical systems eigenvalues equation example Exercise exists finite Fn(x follows function given graph Hence Henon map higher dimensional homeomorphism homoclinic point horizontal hyperbolic set hyperbolic toral automorphism infinitely integer invariant inverse itinerary J(Qc Julia set Lemma linear map Mandelbrot set matrix neighborhood Note open interval parameter period-doubling bifurcation periodic points phase portrait plane point for F points of period polynomial preimage proof Proposition Prove quadratic map remark repelling fixed point repelling periodic points result rotation number Sarkovskii's Theorem satisfies Schwarzian derivative sequence stable and unstable stable set structurally stable symbolic dynamics topological conjugacy topologically conjugate torus unimodal map unique unstable manifolds unstable set vector

### Popular passages

Page 50 - V if 1. / has sensitive dependence on initial conditions. 2. / is topologically transitive. 3. periodic points are dense in V. To summarize, a chaotic map possesses three ingredients: unpredictability, indecomposability, and an element of regularity. A chaotic system is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or decomposed into two subsystems (two invariant open subsets) which do not interact under / because of topological transitivity.

Page 112 - A sequence of real numbers {an} is called a Cauchy sequence if, for any e > 0, there is an integer N such that, for all n, m > N, |an — Om\ < e.

Page 49 - Small errors in computation which are introduced by round-off may become magnified upon iteration. The results of numerical computation of an orbit, no matter how accurate, may bear no resemblance whatsoever with the real orbit.

Page 49 - Intuitively, a map is sensitive to initial conditions, or simply sensitive, if there exist points arbitrarily close to x which eventually separate from x by at least 6 under iteration of F.

Page 49 - J is topologically transitive if it has points which eventually move under iteration from one arbitrarily small neighborhood to any other...

Page 17 - The basic goal of the theory of dynamical systems is to understand the eventual or asymptotic behavior of an iterative process.

Page xiii - Preface to the Second Edition THE RESPONSE to the first edition of...

Page 171 - Y the Cartesian product X x Y is the set of ordered pairs {(x, y): x EX, y GY}.

Page 49 - Intuitively, a topologically transitive map has points which eventually move under iteration from one arbitrarily small neighborhood to any other.

Page 190 - R2 by identifying two points (x, y) and (x', y') if and only if x — x' and y — y