Chaos in dynamical systems / Edward Ott.

Format:

Book

Author/Creator:

Ott, Edward.

Language:

English

Subjects (All):

Chaotic behavior in systems.

Physical Description:

xi, 478 pages : illustrations ; 25 cm

Edition:

Second edition.

Place of Publication:

Cambridge, U.K. ; New York : Cambridge University Press, 2002.

Summary:

New edition of the best-selling graduate textbook on chaos for scientists and engineers.

Contents:

1.1 Some history 1

1.2 Examples of chaotic behavior 2

1.3 Dynamical systems 6

1.4 Attractors 10

1.5 Sensitive dependence on initial conditions 15

1.6 Delay coordinates 19

2 One-dimensional maps 24

2.1 Piecewise linear one-dimensional maps 24

2.2 The logistic map 32

2.3 General discussion of smooth one-dimensional maps 45

2.4 Examples of applications of one-dimensional maps to chaotic systems of higher dimensionality 57

Appendix Some elementary definitions and theorems concerning sets 65

3 Strange attractors and fractal dimension 71

3.1 The box-counting dimension 71

3.2 The generalized baker's map 77

3.3 Measure and the spectrum of D[subscript q] dimensions 80

3.4 Dimension spectrum for the generalized baker's map 84

3.5 Character of the natural measure for the generalized baker's map 85

3.6 The pointwise dimension 89

3.7 Implications and determination of fractal dimension in experiments 91

3.8 A direct experimental observation of fractal attractors 96

3.9 Embedding 98

3.10 Fat fractals 102

Appendix Hausdorff dimension 105

4 Dynamical properties of chaotic systems 115

4.1 The horseshoe map and symbolic dynamics 115

4.2 Linear stability of steady states and periodic orbits 122

4.3 Stable and unstable manifolds 129

4.4 Lyapunov exponents 137

4.5 Entropies 145

4.6 Chaotic flows and magnetic dynamos: the origin of magnetic fields in the Universe 152

Appendix Gram-Schmidt orthogonalization 161

5 Nonattracting chaotic sets 168

5.1 Fractal basin boundaries 169

5.2 Final state sensitivity 175

5.3 Structure of fractal basin boundaries 178

5.4 Chaotic scattering 185

5.5 The dynamics of chaotic scattering 189

5.6 The dimensions of nonattracting chaotic sets and their stable and unstable manifolds 196

5.7 Riddled basins of attraction 201

Appendix Derivation of Eqs. (5.3) 207

6 Quasiperiodicity 212

6.1 Frequency spectrum and attractors 212

6.2 The circle map 218

6.3 N frequency quasiperiodicity with N > 2 228

6.4 Strange nonchaotic attractors of quasiperiodically forced systems 233

6.5 Phase locking of a population of globally coupled oscillators 236

7 Chaos in Hamiltonian systems 246

7.1 Hamiltonian systems 246

7.2 Perturbation of integrable systems 263

7.3 Chaos and KAM tori in systems describable by two-dimensional Hamiltonian maps 273

7.4 Higher-dimensional systems 295

7.5 Strongly chaotic systems 296

7.6 The succession of increasingly random systems 299

8 Chaotic transitions 304

8.1 The period doubling cascade route to chaotic attractors 305

8.2 The intermittency transition to a chaotic attractor 310

8.3 Crises 315

8.4 The Lorenz system: An example of the creation of a chaotic transient 330

8.5 Basin boundary metamorphoses 334

8.6 Bifurcations to chaotic scattering 338

9 Multifractals 345

9.1 The singularity spectrum f([alpha]) 345

9.2 The partition function formalism 353

9.3 Lyapunov partition functions 356

9.4 Distribution of finite time Lyapunov exponents 363

9.5 Unstable periodic orbits and the natural measure 367

9.6 Validity of the Lyapunov and periodic orbits partition functions for nonhyperbolic attractors 371

9.7 Fractal aspects of fluid advection by Lagrangian chaotic flows 373

10 Control and synchronization of chaos 379

10.1 Control of chaos 379

10.2 Controlling a steadily running chaotic process (Goal 1) 381

10.3 Control Goal 2: targeting 390

10.4 Synchronization of chaotic systems 393

10.5 Stability of a chaotic set on an invariant manifold 402

10.6 Generalized synchronization of coupled chaotic systems 409

10.7 Phase synchronization of chaos 411

11 Quantum chaos 421

11.1 The energy level spectra of chaotic, bounded, time-independent systems 423

11.2 Wavefunctions for classically chaotic, bounded, time-independent systems 439

11.3 Temporally periodic systems 442

11.4 Quantum chaotic scattering 449.

Notes:

Includes bibliographical references (pages 452-474) and index.

ISBN:

0521811961

0521010845

OCLC:

48383474

Online:

Publisher description