Dynamical Systems

by Sternberg, Shlomo

• ISBN13: 9780486477053
• ISBN10: 0486477053
• Format: Paperback
• Copyright: 2010-07-21
• Publisher: Dover Publications

Summary

Dynamic systems involve the mathematical expression of any fixed rule describing the time dependence of a point's position in space. This volume features a variety of online components including PowerPoint lecture slides and MATLAB exercises.

Table of Contents

Iteration and fixed points	p. 9
Square roots	p. 9
Analyzing the steps	p. 9
Newton's method	p. 11
A fixed point of the iteration scheme is a solution to our problem	p. 12
The guts of the method	p. 12
A vector version	p. 13
Problems with the implementation of Newton's method	p. 14
The existence theorem	p. 15
Review	p. 18
Basins of attraction	p. 18
Cayley's complex version	p. 20
The implicit function theorem via Newton's method	p. 22
The continuity, differentiability of the implicit function, and the computation of its derivative	p. 23
Attractors and repellers	p. 25
Attractors	p. 25
The basin of attraction of an attractor	p. 25
Repellers	p. 26
Superattractors	p. 26
Notation for iteration	p. 26
Periodic points	p. 26
Renormalization group	p. 27
Iteration for kindergarten	p. 31
Bifurcations	p. 33
The logistic family	p. 33
0 < ¿ ≤ 1	p. 34
¿ = 1	p. 34
¿ > 1	p. 34
1 < ¿ < 2	p. 35
¿ = 2 - the fixed point is superattractive	p. 37
2 < ¿ < 3	p. 37
¿ = 3	p. 40
¿ > 3, points of period two appear	p. 40
3 < ¿ < 1 + &sqrt;6	p. 41
Superattracting period two points	p. 43
1 + &sqrt;6 < ¿	p. 43
Reprise	p. 43
The fold bifurcation	p. 46
The period doubling bifurcation	p. 51
Description of the period doubling bifurcation	p. 51
Statement of the period doubling bifurcation theorem	p. 52
Proof of the period doubling bifurcation theorem	p. 54
Newton's method and Feigenbaum's constant	p. 56
Feigenbaum renormalization	p. 58
Sarkovsky's theorem, Singer's theorem, intermittency	p. 63
Period 3 implies all periods	p. 63
The Sarkovsky ordering	p. 65
Periodic points of period three for the logistic family	p. 65
Singer's theorem	p. 67
The Schwarzian derivative and some of its properties	p. 67
Proof and statement of Singer's theorem	p. 69
Application to the logistic family	p. 70
Intermittency	p. 70
Conjugacy	p. 77
Affine equivalence	p. 77
Conjugacy in general	p. 78
The tent transformation and L4	p. 79
Chaos	p. 81
Transitivity	p. 81
Density of periodic points	p. 83
A definition of chaos	p. 83
The sawtooth transformation and the shift	p. 84
Sensitivity to initial conditions	p. 89
Conjugacy for monotone maps	p. 91
Sequence space and symbolic dynamics	p. 93
A new sequence space	p. 98
The itinerary map	p. 99
Space and time averages	p. 103
Histograms and invariant densities	p. 103
Historgrams of iterations	p. 103
The histogram of L4	p. 107
The mean ergodic theorem	p. 110
The arc sine law	p. 113
The random walk	p. 113
The reflection principle	p. 115
The Beta distributions	p. 121
Two proofs of Stirling's formula	p. 125
The Euler-Maclauren summation formula	p. 125
Euler's integral and Stirling's formula	p. 126
The contraction fixed point theorem	p. 129
Metrics and metric spaces	p. 129
Completeness and completion	p. 133
Normed vector spaces	p. 134
The contraction fixed point theorem	p. 134
Local contractions	p. 135
Dependence on a parameter	p. 136
The Lipschitz implicit function theorem	p. 137
The inverse function theorem	p. 137
The implicit function theorem	p. 139
The local existence theorem for solutions of differential equations	p. 140
The Hausdorff metric and Hutchinson's theorem	p. 143
The Hausdorff metric	p. 143
Contractions and the Hausdorff metric	p. 145
Hutchinson's theorem	p. 145
Affine examples	p. 146
The classical Cantor set	p. 146
The Sierpinski gasket	p. 148
A one line code for creating the Sierpinski gasket	p. 149
Hausdorff dimension	p. 153
Similarity dimension of contracting ratio lists	p. 154
Contracting ratio lists	p. 154
Iterated function systems and fractals	p. 155
Realizations of a contracting ratio list	p. 155
Fractals and fractal dimension	p. 156
Hyperbolicity	p. 159
The conjugacy theorem	p. 159
A global version	p. 160
The local version	p. 163
C∞ conjugacy	p. 165
Invariant manifolds	p. 165
The Lipschitzian case	p. 167
The Perron-Frobenius theorem	p. 175
Non-negative and positive matrices	p. 175
Primitive and irreducible non-negative square matrices	p. 176
Statement of the Perron-Frobenius theorem	p. 176
Proof of the Perron-Frobenius theorem	p. 177
Graphology	p. 181
Non-negative matrices and directed graphs	p. 181
Cycles and primitivity	p. 182
The Frobenius analysis of the irreducible non-primitive case	p. 183
Asymptotic behavior of powers of a primitive matrix	p. 185
The Leslie model of population growth	p. 186
When is the Leslie matrix primitive?	p. 188
The limiting behavior when the Leslie matrix is primitive	p. 188
Markov chains in a nutshell	p. 189
The Google ranking	p. 189
The basic equation	p. 190
Problems with H, the matrix S	p. 190
Problems with S, the Google matrix G	p. 191
Avoiding multiplying by G	p. 192
Eigenvalue sensitivity and reproductive value	p. 193
Some topics in ordinary differential equations	p. 195
Linear equations with constant coefficients	p. 195
Linear homogenous equations with constant coefficients	p. 195
etB where B is a two by two real matrix	p. 197
Hyperbolicity for differential equations	p. 199
Bifurcations of differential equations	p. 199
Variation of constants	p. 199
The operator version	p. 200
The parametrix expansion	p. 201
The Poincaré-Bendixon theorem	p. 202
The ¿-limit set	p. 202
Statement of the Poincaré-Bendixon theorem	p. 203
Properties of the omega limit set of a trajectory, in the general case	p. 203
Proof of Poincaré-Bendixon	p. 205
The van der Pol and Lienard equations	p. 209
The van der Pol equation	p. 209
The Lienard equations	p. 209
Proofs	p. 211
Relaxation oscillations	p. 216
Lotka - Volterra	p. 219
The original Lotka - Volterra equations	p. 219
The null-clines and the zeros	p. 220
Volterra's explanation of why fishing decreases the number of predators	p. 222
A more realistic model	p. 222
Competition between species	p. 225
The n-dimensional Lotka-Volterra equation	p. 228
A theorem of Liapounov	p. 228
Food chains	p. 232
Replicator dynamics and evolutionary stable strategies	p. 234
The replicator equation	p. 234
Linear fitness	p. 234
Hofbauer's equivalence theorem	p. 235
Nash equilibria	p. 236
Evolutionary stable states	p. 237
Entropy and communication	p. 238
Codes	p. 238
Uniquely decipherable codes and instantaneous codes	p. 239
The expected length of a code	p. 239
Shannon's "first theorem"	p. 240
Symbolic dynamics	p. 245
Sequence spaces	p. 245
Exclusions	p. 246
Shifts	p. 246
Homomorphisms between shifts are sliding block codes	p. 247
Shifts of finite type	p. 248
One step shifts	p. 249
Directed multigraphs	p. 249
The adjacency matrix of a directed multigraph	p. 250
The number of fixed points	p. 251
The zeta function	p. 251
Topological entropy	p. 252
Factors of finite shifts	p. 256
The Henon map and symbolic dynamics	p. 257
Bibliography	p. 261
Index	p. 263
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