Chaos and time-series analysis

Title:

Personal Author:

Sprott, Julien C.

Publication Information:

Oxford : Oxford University Press, 2003

ISBN:

9780198508397

Subject Term:

Chaotic behavior in systems

Time-series analysis

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000004610089	Q172.5.C45 S67 2003	Open Access Book	Book	Searching... Unknown

This text provides an introduction to the exciting new developments in chaos and related topics in nonlinear dynamics, including the detection and quantification of chaos in experimental data, fractals, and complex systems. Most of the important elementary concepts in nonlinear dynamics are discussed, with emphasis on the physical concepts and useful results rather than mathematical proofs and derivations. While many books on chaos are purely qualitative and many others are highly mathematical, this book fills the middle ground by giving the essential equations, but in the simplest possible form. It assumes only an elementary knowledge of calculus. Complex numbers, differential equations, and vector calculus are used in places, but those tools are described as required. The book is aimed at the student, scientist, or engineer who wants to learn how to use the ideas in a practical setting. It is written at a level suitable for advanced undergraduate and beginning graduate students in all fields of science and engineering.

Author Notes

Professor Julien Clinton SprottDepartment of PhysicsUniversity of Wisconsin-Madison1150 University AvenueMadisonWisconsin 53706USATel: 001-608-263-4449Email: sprott@physics.wisc.eduhttp://sprott.physics.wisc.edu/

Reviews 1

Choice Review

Sprott (physics, Univ. of Wisconsin-Madison) analyzes "chaotic" physical examples, e.g., dynamical systems. He discusses "chaos" using common examples, like dripping faucets, electrical circuits, and fluids, as well as discrete dynamical systems through one-dimensional maps, especially logistics. Sprott returns to these initial ideas, both mathematical concepts and examples, throughout; e.g., he returns to one-dimensional mappings to introduce Lyapunov exponents. He takes readers through continuous examples using both one-dimensional flows and systems of nonlinear differential equations, such as Hamiltonian systems (like the undamped pendulum and the driven pendulum). He looks at numerical methods as they relate to "chaotic" examples; investigates the dynamical nature of Newton's method; and studies attractors and strange attractors. Different types of bifurcations are examined for both maps and flows. He introduces time-series through several examples and nonlinear analysis, and how it varies from more typical linear analysis. Other chapters treat fractals, with examples like Hilbert's cube, Koch snowflakes, Julia sets, Sierpinski triangles, and Cantor sets and how to calculate the fractional dimension of these objects; and spatiotemporal chaos--how interactions between objects on a local level affects the whole system. Essential equations and mathematics help explain points. Problems; chapter computer programming projects; comparison tables; appendix of common "chaotic" systems. ^BSumming Up: Recommended. Upper-division undergraduates through professionals. M. D. Sanford St. Francis University

1 Introduction	p. 1
1.1 Examples of dynamical systems	p. 2
1.2 Driven pendulum	p. 5
1.3 Ball on an oscillating floor	p. 8
1.4 Dripping faucet	p. 9
1.5 Chaotic electrical circuits	p. 11
1.6 Other demonstrations of chaos	p. 16
1.7 Exercises	p. 18
1.8 Computer project: The logistic equation	p. 19
2 One-dimensional maps	p. 20
2.1 Exponential growth in discrete time	p. 20
2.2 The logistic equation	p. 22
2.3 Bifurcations in the logistic map	p. 24
2.4 Other properties of the logistic map with A = 4	p. 31
2.5 Other one-dimensional maps	p. 36
2.6 Computer random-number generators	p. 41
2.7 Exercises	p. 43
2.8 Computer project: Bifurcation diagrams	p. 45
3 Nonchaotic multi-dimensional flows	p. 46
3.1 Exponential growth in continuous time	p. 46
3.2 Logistic differential equation	p. 48
3.3 Circular motion	p. 50
3.4 Simple harmonic oscillator	p. 52
3.5 Driven harmonic oscillator	p. 53
3.6 Damped harmonic oscillator	p. 57
3.7 Driven damped harmonic oscillator	p. 59
3.8 Van der Pol equation	p. 61
3.9 Numerical solution of differential equations	p. 63
3.10 Exercises	p. 68
3.11 Computer project: Van der Pol equation	p. 70
4 Dynamical systems theory	p. 72
4.1 Two-dimensional equilibria	p. 72
4.2 Stability of two-dimensional equilibria	p. 74
4.3 Damped harmonic oscillator revisited	p. 75
4.4 Saddle points	p. 76
4.5 Area contraction and expansion	p. 78
4.6 Nonchaotic three-dimensional attractors	p. 80
4.7 Stability of two-dimensional maps	p. 85
4.8 Chaotic dissipative flows	p. 86
4.9 Shadowing	p. 99
4.10 Exercises	p. 100
4.11 Computer project: The Lorenz attractor	p. 102
5 Lyapunov exponents	p. 104
5.1 Lyapunov exponent for one-dimensional maps	p. 105
5.2 Lyapunov exponents for two-dimensional maps	p. 110
5.3 Lyapunov exponent for one-dimensional flows	p. 113
5.4 Lyapunov exponents for two-dimensional flows	p. 114
5.5 Lyapunov exponents for three-dimensional flows	p. 115
5.6 Numerical calculation of the largest Lyapunov exponent	p. 116
5.7 Lyapunov exponent spectrum in arbitrary dimension	p. 118
5.8 General characteristics of Lyapunov exponents	p. 119
5.9 Kaplan-Yorke (or Lyapunov) dimension	p. 121
5.10 Precautions	p. 122
5.11 Exercises	p. 123
5.12 Computer project: Lyapunov exponent	p. 125
6 Strange attractors	p. 127
6.1 General properties	p. 127
6.2 Examples	p. 129
6.3 Search methods	p. 131
6.4 Probability of chaos	p. 135
6.5 Statistical properties	p. 137
6.6 Visualization methods	p. 141
6.7 Unstable periodic orbits	p. 146
6.8 Basins of attraction	p. 147
6.9 Structural stability and robustness	p. 151
6.10 Aesthetics	p. 153
6.11 Exercises	p. 156
6.12 Computer project: Henon map	p. 158
7 Bifurcations	p. 159
7.1 Bifurcations in one-dimensional flows	p. 160
7.2 Hopf bifurcation	p. 164
7.3 Bifurcations in one-dimensional maps	p. 166
7.4 Neimark-Sacker bifurcation	p. 169
7.5 Homoclinic and heteroclinic bifurcations	p. 170
7.6 Crises	p. 175
7.7 Exercises	p. 180
7.8 Computer project: Poincare sections	p. 182
8 Hamiltonian chaos	p. 184
8.1 Mass on a spring	p. 185
8.2 Hamilton's equations	p. 185
8.3 Properties of Hamiltonian systems	p. 186
8.4 Simple pendulum	p. 187
8.5 Driven pendulum	p. 190
8.6 Other driven nonlinear oscillators	p. 192
8.7 Henon-Heiles system	p. 194
8.8 Three-dimensional conservative flows	p. 195
8.9 Symplectic maps	p. 199
8.10 KAM theory	p. 206
8.11 Exercises	p. 207
8.12 Computer project: Chirikov map	p. 209
9 Time-series properties	p. 211
9.1 Hierarchy of dynamical behaviors	p. 212
9.2 Examples of experimental time series	p. 213
9.3 Practical considerations	p. 215
9.4 Conventional linear methods	p. 216
9.5 Case study	p. 228
9.6 Time-delay embeddings	p. 236
9.7 Summary of important dimensions	p. 238
9.8 Exercises	p. 239
9.9 Computer project: Autocorrelation function	p. 241
10 Nonlinear prediction and noise reduction	p. 243
10.1 Linear predictors	p. 244
10.2 State-space prediction	p. 247
10.3 Noise reduction	p. 251
10.4 Lyapunov exponents from experimental data	p. 253
10.5 False nearest neighbors	p. 256
10.6 Principal component analysis	p. 260
10.7 Artificial neural network predictors	p. 263
10.8 Exercises	p. 269
10.9 Computer project: Nonlinear prediction	p. 271
11 Fractals	p. 273
11.1 Cantor sets	p. 274
11.2 Fractal curves	p. 276
11.3 Fractal trees	p. 282
11.4 Fractal gaskets	p. 284
11.5 Fractal sponges	p. 287
11.6 Random fractals	p. 288
11.7 Fractal landscapes	p. 294
11.8 Natural fractals	p. 297
11.9 Exercises	p. 299
11.10 Computer project: State-space reconstruction	p. 300
12 Calculation of the fractal dimension	p. 302
12.1 Similarity dimension	p. 303
12.2 Capacity dimension	p. 304
12.3 Correlation dimension	p. 307
12.4 Entropy	p. 311
12.5 BDS statistic	p. 313
12.6 Minimum mutual information	p. 314
12.7 Practical considerations	p. 317
12.8 Fractal dimension of graphic images	p. 323
12.9 Exercises	p. 326
12.10 Computer project: Correlation dimension	p. 327
13 Fractal measure and multifractals	p. 329
13.1 Convergence of the correlation dimension	p. 330
13.2 Multifractals	p. 336
13.3 Examples of generalized dimensions	p. 340
13.4 Numerical calculation of generalized dimensions	p. 341
13.5 Singularity spectrum	p. 343
13.6 Generalized entropies	p. 345
13.7 Unbounded strange attractors	p. 346
13.8 Summary of time-series analysis methods	p. 348
13.9 Exercises	p. 349
13.10 Computer project: Iterated function systems	p. 351
14 Nonchaotic fractal sets	p. 353
14.1 The chaos game	p. 353
14.2 Affine transformations	p. 356
14.3 Iterated function systems	p. 358
14.4 Julia sets	p. 364
14.5 The Mandelbrot set	p. 368
14.6 Generalized Julia sets	p. 372
14.7 Basins of Newton's method	p. 374
14.8 Computational considerations	p. 376
14.9 Exercises	p. 377
14.10 Computer project: Mandelbrot and Julia sets	p. 379
15 Spatiotemporal chaos and complexity	p. 381
15.1 Cellular automata	p. 382
15.2 Self-organized criticality	p. 390
15.3 The Ising model	p. 394
15.4 Percolation	p. 396
15.5 Coupled lattices	p. 398
15.6 Infinite-dimensional systems	p. 401
15.7 Measures of complexity	p. 410
15.8 Summary of spatiotemporal models	p. 412
15.9 Concluding remarks	p. 413
15.10 Exercises	p. 413
15.11 Computer project: Spatiotemporal chaos and complexity	p. 416
A Common chaotic systems	p. 417
A.1 Noninvertible maps	p. 417
A.2 Dissipative maps	p. 421
A.3 Conservative maps	p. 425
A.4 Driven dissipative flows	p. 428
A.5 Autonomous dissipative flows	p. 431
A.6 Conservative flows	p. 439
B Useful mathematical formulas	p. 441
B.1 Trigonometric relations	p. 441
B.2 Hyperbolic functions	p. 441
B.3 Logarithms	p. 442
B.4 Complex numbers	p. 442
B.5 Derivatives	p. 443
B.6 Integrals	p. 443
B.7 Approximations	p. 444
B.8 Matrices and determinants	p. 444
B.9 Roots of polynomials	p. 445
B.10 Vector calculus	p. 446
C Journals with chaos and related papers	p. 447
Bibliography	p. 449
Index	p. 485

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