Deterministic Chaos

Title page
Preface
Color Plates
1 Introduction
2 Experiments and Simple Models
2.1 Experimental Detection of Deterministic Chaos
2.1.1 Driven Pendulum
2.1.2 Rayleigh-Bénard System in a Box
2.1.3 Stirred Chemical Reactions
2.1.4 Hénon-Heiles System
2.2 The Periodically Kicked Rotator
2.2.1 Logistic Map
2.2.2 Hénon Map
2.2.3 Chirikov Map
3 Piecewise Linear Maps and Deterministic Chaos
3.1 The Bemoulli Shift
3.2 Characterization of Chaotic Motion
3.2.1 Liapunov Exponent
3.2.2 Invariant Measure
3.2.3 Correlation Function
3.3 Deterministic Diffusion
4 Universal Behavior of Quadratic Maps
4.1 Parameter Dependence of the Iterates
4.2 Pitchfork Bifurcation and the Doubling Transformation
4.2.1 Pitchfork Bifurcations
4.2.2 Supercycles
4.2.3 Doubling Transformation and α
4.2.4 Linearized Doubling Transformation and δ
4.3 Self-Similarity, Universal Power Spectrum, and the Influence of Extemal Noise
4.3.1 Self-Similarity in the Positions of the Cycle Elements
4.3.2 Hausdorff Dimension
4.3.3 Power Spectrum
4.3.4 Influence of Extemal Noise
4.4 Behavior of the Logistic Map for r_∞ty < r
4.4.1 Sensitive Dependence on Parameters
4.4.2 Structural Universality
4.4.3 Chaotic Bands and Scaling
4.5 Parallels between Period Doubling and Phase Transitions
4.6 Experimental Support for the Bifurcation Route
5 The Intermittency Route to Chaos
5.1 Mechanisms for Intermittency
5.1.1 Type-I Intermittency
5.1.2 Length of the Laminar Region
5.2 Renormalization-Group Treatment of Intermittency
5.3 Intermittency and l/f-Noise
5.4 Experimental Observation of the Intermittency Route
5.4.1 Distribution of Laminar Lengths
5.4.2 Type-I Intermittency
5.4.3 Type-III Intermittency
6 Strange Attractors in Dissipative Dynamical Systems
6.1 Introduction and Definition of Strange Attractors
6.1.1 Baker's Transformation
6.1.2 Dissipative Hénon Map
6.2 The Kolmogorov Entropy
6.2.1 Definition of K
6.2.2 Connection of K to the Liapunov Exponents
6.2.3 Average Time over which the State of a Chaotic System can be Predicted l00
6.3 Characterization of the Attractor by a Measured Signal
6.3.1 Reconstruction of the Attractor from a Time Series
6.3.2 Generalized Dimensions and Distribution of Singularities in the Invariant Density
6.3.3 Generalized Entropies and Fluctuations around the K-Entropy
6.3.4 Kaplan-Yorke Conjecture
6.4 Pictures of Strange Attractors and Fractal Boundaries
7 The Transition from Quasiperiodicity to Chaos
7.1 Strange Attractors and the Onset of Turbulence
7.1.1 Hopf Bifurcation
7.1.2 Landau's Route to Turbulence
7.1.3 Ruelle-Takens-Newhouse Route to Chaos
7.1.4 Possibility of Three-Frequency Quasiperiodic Orbits
7.1.5 Break-up of a Two-Torus
7.2 Universal Properties of the Transition from Quasiperiodicity to Chaos
7.2.1 Mode Locking and the Farey Tree
7.2.2 Local Universality
7.2.3 Global Universality
7.3 Experiments and Circle Maps
7.3.1 Driven Pendulum
7.3.2 Electrical Conductivity in Barium Sodium Niobate
7.3.3 Dynamics of Cardiac Cells
7.3.4 Forced Rayleigh-Bénard Experiment
7.4 Routes to Chaos
7.4.1 Crises
8 Regular and Irregular Motion in Conservative Systems
8.1 Coexistence of Regular and Irregular Motion
8.1.1 Integrable Systems
8.1.2 Perturbation Theory and Vanishing Denominators
8.1.3 Stable Tori and KAM Theorem
8.1.4 Unstable Tori and Poincaré-Birkhoff Theorem
8.1.5 Homoclinic Points and Chaos
8.1.6 Arnold Diffusion
8.1.7 Examples of Classical Chaos
8.2 Strangly Irregular Motion and Ergodicity
8.2.1 Cat Map
8.2.2 Hierarchy of Classical Chaos
8.2.3 Three Classical K-Systems
9 Chaos in Quantum Systems?
9.1 The Quantum Cat Map
9.2 A Quantum Particle in a Stadium
9.3 The Kicked Quantum Rotator
10 Controlling Chaos
10.1 Stabilization of Unstable Orbits
10.2 The OGY Method
10.3 Time-Delayed Feedback Contrai
10.3.1 Rhythmic Contra!
10.3.2 Extended Time-Delayed Feedback Control
10.3.3 Experimental Realization of Time-Delayed Feedback Control
10.4 Parametric Resonance from Unstable Periodic Orbits
11 Synchronization of Chaotic Systems
11.1 Identical Systems with Symmetric Coupling
11.1.1 On-Off Intermittency
11.1.2 Strang ys. Weak Synchronization
11.2 Master-Slave Configurations
11.3 Generalized Synchronization
11.3.1 Strange Nonchaotic Attractors
11.4 Phase Synchronization of Chaotic Systems
12 Spatiotemporal Chaos
12.1 Models for Space-Time Chaos
12.1.1 Coupled Map Lattices
12.1.2 Coupled Oscillator Models
12.1.3 Complex Ginzburg-Landau Equation
12.1.4 Kuramoto-Sivashinsky Equation
12.2 Characterization of Space-Time Chaos
12.2.1 Liapunov Spectrum
12.2.2 Co-moving Liapunov Exponent
12.2.3 Chronotopic Liapunov Analysis
12.3 Nonlinear Nonequilibrium Space-Time Dynamics
12.3.1 Fully Developed Turbulence
12.3.2 Spatiotemporal Intermittency
12.3.3 Molecular Dynamics
Outlook
Appendix
A Derivation of the Lorenz Model
B Stability Analysis and the Onset of Convection and Turbulence in the Lorenz Model
C The Schwarzian Derivative
D Renormalization of the One-Dimensional Ising Model
E Decimation and Path Integrals for Extemal Noise
F Shannon's Measure of Information
F.l Information Capacity of a Store
F.2 Information Gain
G Period Doubling for the Conservative Hénon Map
H Unstable Periodic Orbits
Remarks and References
Index