An Introduction to Chaotic Dynamical Systems

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Westview Press, 2003 - Science - 335 pages
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The 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

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
Copyright

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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

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About the author (2003)

Professor Robert L. Devaney received his A.B. from Holy Cross College and his Ph.D. from the University of California at Berkeley in 1973. He taught at Northwestern University, Tufts University, and the University of Maryland before coming to Boston University in 1980. He served there as chairman of the Department of Mathematics from 1983 to 1986. His main area of research is dynamical systems, including Hamiltonian systems, complex analytic dynamics, and computer experiments in dynamics. He is the author of An Introduction to Chaotic Dynamical Systems, and Chaos, Fractals, and Dynamics: Computer Experiments in Modern Mathematics, which aims to explain the beauty of chaotic dynamics to high school students and teachers.

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