Invitation to Dynamical Systems

By: EDWARD R SCHEINERMAN

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Paperback | 15 February 2012

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408 Pages

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Designed for those wishing to study mathematics beyond linear algebra but unready for abstract material, this "invitation" to the excitement of dynamical systems appeals to readers from a wide range of backgrounds. Rather than taking a theorem-proof-corollary-remark approach, it stresses geometry and intuition. Topics include both the classical theory of linear systems and the modern theory of nonlinear and chaotic systems as well as bifurcation, symbolic dynamics, fractals, and complex systems.
In addition to offering a unified presentation of continuous and discrete time systems, this treatment integrates computing comfortably into the text. Appendixes feature important background material, including a gentle introduction to differential equations and explanations of how to write MATLAB, Mathematica, and C programs to compute dynamical systems. Prerequisites for advanced undergraduates and graduate students include two semesters of calculus and one semester of linear algebra.

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Geometry Applied Mathematics Chaos Theory Science Science in General Physics Classical Mathematics Dynamics & Statics

Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory

Preface	p. XI
Introduction	p. i
What is a dynamical system?	p. 1
State vectors	p. 1
The next instant: discrete time	p. 2
The next instant: continuous time	p. 4
Summary	p. 6
Problems	p. 6
Examples	p. 8
Mass and spring	p. 8
RLC circuits	p. 10
Pendulum	p. 11
Your bank account	p. 16
Economic growth	p. 17
Pushing buttons on your calculator	p. 19
Microbes	p. 22
Predator and prey	p. 24
Newton's Method	p. 26
Euler's method	p. 28
"Random" number generation	p. 31
Problems	p. 32
What we want; what we can get	p. 35
Linear Systems	p. 37
One dimension	p. 37
Discrete time	p. 37
Continuous time	p. 45
Summary	p. 48
Problems	p. 49
Two (and more) dimensions	p. 50
Discrete time	p. 51
Continuous time	p. 57
The nondiagonalizable case*	p. 81
Problems	p. 88
Examplification: Markov chains	p. 91
Introduction	p. 91
Markov chains as linear systems	p. 93
The long term	p. 96
Problems	p. 98
Nonlinear Systems 1: Fixed Points	p. 101
Fixed points	p. 102
What is a fixed point?	p. 102
Finding fixed points	p. 103
Stability	p. 104
Problems	p. 108
Linearization	p. 110
One dimension	p. 110
Two and more dimensions	p. 117
Problems	p. 126
Lyapunov functions	p. 128
Linearization can fail	p. 128
Energy	p. 130
Lyapunov's method	p. 133
Gradient systems	p. 137
Problems	p. 144
Examplification: Iterative methods for solving equations	p. 146
Problems	p. 150
Nonlinear Systems 2: Periodicity and Chaos	p. 153
Continuous time	p. 154
One dimension: no periodicity	p. 154
Two dimensions: the Poincare-Bendixson theorem	p. 155
The Hopfbifurcation	p. 161
Higher dimensions: the Lorenz system and chaos	p. 163
Problems	p. 167
Discrete time	p. 168
Periodicity	p. 169
stability of periodic points	p. 173
Bifurcation	p. 175
Sarkovskii's theorem	p. 188
Chaos and symbolic dynamics	p. 202
Problems	p. 216
Examplification: Riffle shuffles and the shift map	p. 218
Riffle shuffles	p. 218
The shift map	p. 219
Shifting and shuffling	p. 223
Shuffling again and again	p. 226
Problems	p. 229
Fractals	p. 231
Cantor's set	p. 231
Symbolic representation of Cantor's set	p. 232
Cantor's set in conventional notation	p. 233
The link between the two representations	p. 236
Topological properties of the Cantor set	p. 236
In what sense a fractal?	p. 240
Problems	p. 241
Biting out the middle in the plane	p. 242
Sierpiriski's triangle	p. 242
Koch's snowflake	p. 243
Problems	p. 245
Contraction mapping theorems	p. 246
Contraction maps	p. 246
Contraction mapping theorem on the real line	p. 247
Contraction mapping in higher dimensions	p. 249
Contractive affine maps: the spectral norm*	p. 250
Other metric spaces	p. 254
Compact sets and Hausdorff distance	p. 255
Problems	p. 258
Iterated function systems	p. 260
From point maps to set maps	p. 260
The union of set maps	p. 262
Examples revisited	p. 265
IFSs defined	p. 270
Working backward	p. 271
Problems	p. 276
Algorithms for drawing fractals	p. 276
A deterministic algorithm	p. 277
Dancing on fractals279
A randomized algorithm	p. 283
Problems	p. 286
Fractal dimension	p. 287
Covering with balls	p. 287
Definition of dimension	p. 290
Simplifying the definition	p. 292
Just-touching similitudes and dimension	p. 300
Problems	p. 306
Examplification: Fractals in nature	p. 307
Dimension of physical fractals	p. 308
Estimating surface area	p. 312
Image analysis	p. 314
Problems	p. 314
Complex Dynamical Systems	p. 317
Julia sets	p. 317
Definition and examples	p. 317
Escape-time algorithm	p. 324
Other Julia sets	p. 326
Problems	p. 327
The Mandelbrot set	p. 327
Definition and various views	p. 327
Escape-time algorithm	p. 332
Problems	p. 332
Examplification: Newton's method revisited	p. 333
Problems	p. 336
Examplification: Complex bases	p. 336
Place value revisited	p. 336
ifss revisited	p. 338
Problems	p. 340
Background Material	p. 341
Linear algebra	p. 341
Much ado about 0	p. 341
Linear independence	p. 342
Eigenvalues/vectors	p. 342
agonalization	p. 343
Jordan canonical form	p. 343
Basic linear transformations of the plane	p. 344
Complex numbers	p. 347
Calculus	p. 349
Intermediate and mean value theorems	p. 349
Partial derivatives	p. 350
Differential equations	p. 351
Equations	p. 351
What is a differential equation?	p. 351
Standard notation	p. 352
Computing	p. 355
Differential equations	p. 355
Analytic solutions	p. 356
Numerical solutions	p. 357
Triangle Dance	p. 365
About the accompanying software	p. 366
Bibliography	p. 369
Index	p. 371
Table of Contents provided by Ingram. All Rights Reserved.

Invitation to Dynamical Systems

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