Chaos in Dynamical Systems 2nd Edition by Edward Ott 0521010845 9780521010849

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ISBN 10: 0521010845

ISBN 13: 9780521010849

Author: Edward Ott

Over the past two decades scientists, mathematicians, and engineers have come to understand that a large variety of systems exhibit complicated evolution with time. This complicated behavior is known as chaos. In the new edition of this classic textbook Edward Ott has added much new material and has significantly increased the number of homework problems. The most important change is the addition of a completely new chapter on control and synchronization of chaos. Other changes include new material on riddled basins of attraction, phase locking of globally coupled oscillators, fractal aspects of fluid advection by Lagrangian chaotic flows, magnetic dynamos, and strange nonchaotic attractors. This new edition will be of interest to advanced undergraduates and graduate students in science, engineering, and mathematics taking courses in chaotic dynamics, as well as to researchers in the subject.

Chaos in Dynamical Systems 2nd Table of contents:

1 Introduction and overview

1.1 Some history

1.2 Examples of chaotic behavior

1.3 Dynamical systems

1.4 Attractors

1.5 Sensitive dependence on initial conditions

1.6 Delay coordinates

Problems

Notes

2 One–dimensional maps

2.1 Piecewise linear one–dimensional maps

2.2 The logistic map

2.3 General discussion of smooth one–dimensional maps

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

Appendix: Some elementary definitions and theorems concerning sets

Problems

Notes

3 Strange attractors and fractal dimension

3.1 The box–counting dimension

3.2 The generalized baker’s map

3.3 Measure and the spectrum of D[sub(q)] dimensions

3.4 Dimension spectrum for the generalized baker’s map

3.5 Character of the natural measure for the generalized baker’s map

3.6 The pointwise dimension

3.7 Implications and determination of fractal dimension in experiments

3.8 A direct experimental observation of fractal attractors

3.9 Embedding

3.10 Fat fractals

Appendix: Hausdorff dimension

Problems

Notes

4 Dynamical properties of chaotic systems

4.1 The horseshoe map and symbolic dynamics

4.2 Linear stability of steady states and periodic orbits

4.3 Stable and unstable manifolds

4.4 Lyapunov exponents

4.5 Entropies

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

Appendix: Gram–Schmidt orthogonalization

Problems

Notes

5 Nonattracting chaotic sets

5.1 Fractal basin boundaries

5.2 Final state sensitivity

5.3 Structure of fractal basin boundaries

5.4 Chaotic scattering

5.5 The dynamics of chaotic scattering

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

5.7 Riddled basins of attraction

Appendix: Derivation of Eqs. (5.3)

Problems

Notes

6 Quasiperiodicity

6.1 Frequency spectrum and attractors

6.2 The circle map

6.3 Nfrequency quasiperiodicity with N > 2

6.4 Strange nonchaotic attractors of quasiperiodically forced systems

6.5 Phase locking of a population of globally coupled oscillators

Problems

Notes

7 Chaos in Hamiltonian systems

7.1 Hamiltonian systems

7.2 Perturbation of integrable systems

7.3 Chaos and KAM tori in systems describable by twodimensional Hamiltonian maps

7.4 Higher–dimensional systems

7.5 Strongly chaotic systems

7.6 The succession of increasingly random systems

Problems

Notes

8 Chaotic transitions

8.1 The period doubling cascade route to chaotic attractors

8.2 The intermittency transition to a chaotic attractor

8.3 Crises

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

8.5 Basin boundary metamorphoses

8.6 Bifurcations to chaotic scattering

Problems

Notes

9 Multifractals

9.1 The singularity spectrum f(α)

9.2 The partition function formalism

9.3 Lyapunov partition functions

9.4 Distribution of finite time Lyapunov exponents

9.5 Unstable periodic orbits and the natural measure

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

9.7 Fractal aspects of fluid advection by Lagrangian chaotic flows

Problems

Notes

10 Control and synchronization of chaos

10.1 Control of chaos

10.2 Controlling a steadily running chaotic process (Goal 1)

10.3 Control Goal 2: targeting

10.4 Synchronization of chaotic systems

10.5 Stability of a chaotic set on an invariant manifold

10.6 Generalized synchronization of coupled chaotic systems

10.7 Phase synchronization of chaos

Problems

Notes

11 Quantum chaos

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

11.2 Wavefunctions for classically chaotic, bounded, time–independent systems

11.3 Temporally periodic systems

11.4 Quantum chaotic scattering

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