Chaos: A Mathematical Introduction

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Cambridge University Press, May 8, 2003 - Mathematics - 294 pages
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When new ideas like chaos first move into the mathematical limelight, the early textbooks tend to be very difficult. The concepts are new and it takes time to find ways to present them in a form digestible to the average student. This process may take a generation, but eventually, what originally seemed far too advanced for all but the most mathematically sophisticated becomes accessible to a much wider readership. This book takes some major steps along that path of generational change. It presents ideas about chaos in discrete time dynamics in a form where they should be accessible to anyone who has taken a first course in undergraduate calculus. More remarkably, it manages to do so without discarding a commitment to mathematical substance and rigour. The book evolved from a very popular one-semester middle level undergraduate course over a period of several years and has therefore been well class-tested.
 

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Contents

MAKING PREDICTIONS
1
11 MATHEMATICAL MODELS
2
12 CONTINUOUS GROWTH MODELS
7
13 DISCRETE GROWTH MODELS
11
14 NUMERICAL SOLUTIONS
17
15 DYNAMICAL SYSTEMS
20
MAPPINGS AND ORBITS
27
21 MAPPINGS
28
81 DIVERGING ITERATES
142
82 DEFINITION OF SENSITIVITY
144
83 USING THE DEFINITION
149
84 SENSITIVITY AND CHAOS
155
INGREDIENTS OF CHAOS
157
91 SENSITIVITY EVERYWHERE
158
92 DENSE ORBITS AND TRANSITIVITY
161
93 DENSE PERIODIC POINTS
166

22 TIME SERIES
33
23 ORBITS
38
PERIODIC ORBITS
45
31 FIXED POINTS AND PERIODIC POINTS
46
32 FINDING FIXED POINTS
50
33 EVENTUALLY PERIODIC ORBITS
57
ASYMPTOTIC ORBITS I LINEAR AND AFFINE MAPPINGS
61
41 COBWEB DIAGRAMS
62
42 LINEAR MAPPINGS
66
43 AFFINE MAPPINGS
70
ASYMPTOTIC ORBITS II DIFFERENTIABLE MAPPINGS
75
51 DIFFERENTIABLE MAPPINGS
76
52 MAIN THEOREM AND PROOF
83
53 PERIODIC POINTS
86
FAMILIES OF MAPPINGS AND BIFURCATIONS
91
61 THREE FAMILIES OF MAPPINGS
92
62 FAMILIES OF FIXED POINTS
98
63 FAMILIES OF PERIOD 2 POINTS
106
64 THE BIFURCATION DIAGRAM
113
GRAPHICAL COMPOSITION WIGGLY ITERATES AND ZEROS
117
71 GRAPHING THE SECOND ITERATE
118
72 ONEHUMP MAPPINGS
125
73 DENSE SETS OF ZEROES
132
74 ZEROES UNDER BACKWARD ITERATION
135
SENSITIVE DEPENDENCE
141
94 WHAT IS CHAOS?
168
95 CONVERSE LEMMAS
170
SCHWARZIAN DERIVATIVE AND WOGGLES
179
101 WIGGLES AND WOGGLES
180
102 THE SCHWARZIAN DERIVATIVE
186
103 TESTING FOR CHAOS
191
104 PROVING THEOREM 1031
193
CHANGING COORDINATES
203
111 CHANGE OF VARIABLE
204
112 COORDINATES
207
113 MAPPINGS ON THE LINE
211
CONJUGACY
217
121 DEFINITION AND EXAMPLES
218
122 APPROXIMATING A CONJUGACY
225
123 EXISTENCE OF A CONJUGACY
238
124 CONJUGACY AND DYNAMICS
247
WIGGLY ITERATES CANTOR SETS AND CHAOS
255
131 ITERATING ONEHUMP MAPPINGS
256
132 CONSTRUCTING AN INVARIANT SET
265
133 WIGGLY ITERATES AND CANTOR SETS
272
134 DENSE IN A CANTOR SET
278
135 CHAOS ON CANTOR SETS
282
136 TESTS FOR CHAOS AND CONJUGACY
284
INDEX
287
Copyright

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About the author (2003)

John Banks taught school for 18 years before becoming a full-time writer. He has written over 200 books about tornadoes, orangutans, soccer, space travel, and other topics. In his free time, John enjoys taking his granddaughter, Sarah, on adventures, climbing nearby mountains, and visiting city graveyards. He lives in Hampshire, England, with his wife, Helen, who is also a writer.

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