Preface
Contents
Contributors
How Did You Get into Chaos?
Singular Perturbations of Complex Analytic Dynamical Systems
Robert L. Devaney
1 Introduction
2 Preliminaries
3 The Escape Trichotomy
4 Proof of the Escape Trichotomy
5 Classification of Escape Time Julia Sets
6 Structure Around the McMullen Domain
7 Cantor Necklaces
8 The Case n=2
9 Julia Sets Converging to the Unit Disk
References
Heteroclinic Switching in Coupled Oscillator Networks: Dynamics on Odd Graphs
Peter Ashwin, Gábor Orosz, and Jon Borresen
1 Introduction
2 Dynamics and Bifurcations with SN Symmetry
3 Bifurcations for Three and Four GloballyCoupled Oscillators
4 Heteroclinic Networks for Odd Numbers of Oscillators
4.1 Existence, Stability and Connections of [k,k+1] Cluster States
5 Discussion
References
Dynamics of Finite-Size Particles in Chaotic Fluid Flows
Julyan H.E. Cartwright, Ulrike Feudel, György Károlyi, Alessandro de Moura, Oreste Piro, and Tamás Tél
1 Introduction and Overview
2 Motion of Finite-Size Particles in Fluid Flows
2.1 The Maxey-Riley Equation
2.2 General Features of the Dynamics of Finite-Size Particles
3 Chaotic Advection of Passive Tracers
3.1 Properties of Passive-Tracer Chaotic Advection
3.2 The Convection and Cellular Flow Models
3.3 The Von Kármán Vortex Street
4 Inertial Effects in Closed Chaotic Flows
4.1 Neutrally Buoyant Particles
4.2 3D Flows and Bailout Embeddings
5 Advection of Finite-Size Particles in Open Flows
6 Coagulation and Fragmentation of Finite-Size Particles
7 Future Directions
References
Langevin Equation for Slow Degrees of Freedom of Hamiltonian Systems
R.S. MacKay
1 Introduction
2 Assumptions
3 Aim
4 Strategy
4.1 Zeroth Order Mean Velocity
4.2 Fluctuations
4.3 Correction to Ergode
4.4 Effect of Autonomous Slow Motion
4.5 Ergode to Monode
4.6 Klimontovich Interpretation
5 Case of Standard Mechanical System
6 Quantum Degrees of Freedom
7 Kinetics Out of Chemical Equilibrium
8 Conclusion and Problems
References
Stable Chaos
Antonio Politi and Alessandro Torcini
1 Introduction
2 Models
3 Definition and Characterization of Stable Chaos
4 Relationship with Cellular Automata
5 Relationship with Deterministic Chaos
6 From Order to Chaos
7 More Realistic Models
7.1 A Hamiltonian Model: Diatomic Hard-Point Chain
7.2 Neural Networks
8 Conclusions
References
Superpersistent Chaotic Transients
Ying-Cheng Lai
1 Introduction
2 Unstable -- Unstable Pair Bifurcation
3 Riddling Bifurcation and Superpersistent Chaotic Transients
4 Superpersistent Chaotic Transients in Spatiotemporal Systems
5 Noise-Induced Superpersistent Chaotic Transients
6 Application: Advection of Inertial Particles in Open Chaotic Flows
7 Conclusions
References
Synchronization in Climate Dynamics and Other Extended Systems
Peter L. Read and Alfonso A. Castrejón-Pita
1 Introduction
2 Climate Cycles and Teleconnections
2.1 Cyclic Variations in Climate Variables
2.2 Teleconnections
3 Models and Mechanisms for Teleconnection and Synchronization
3.1 Distinguishing Synchronized Models from Observations?
3.2 Zonally Symmetric Coupling
4 Laboratory Analogues of Zonally-Symmetric Synchronization
4.1 Periodic Perturbations
4.2 Mutual Synchronization Experiments
5 Discussion
References
Stochastic Synchronization
Ram Ramaswamy, R.K. Brojen Singh, Changsong Zhou, and Jürgen Kurths
1 Introduction
2 Measures for Stochastic Synchronization
3 The Effect of Stochasticity on Synchrony
4 The Emergence of Synchrony in Stochastic Systems
5 Discussion and Summary
References
Experimental Huygens Synchronization of Oscillators
Alexander Pogromsky, David Rijlaarsdam, and Henk Nijmeijer
1 Introduction
2 Synchronization of Pendulum Clocks
3 The Goal of the Experimental Set-Up
4 The Experimental Set-Up
4.1 Adjustment of the System Properties
5 Example 1: Coupled Duffing Oscillators
5.1 Problem Statement and Analysis
5.2 Experimental and Numerical Results
6 Example 2: Two Coupled Rotary Disks
6.1 Problem Statement
6.2 Experimental Results
7 Conclusions
References
Controlling Chaos: The OGY Method, Its Use in Mechanics, and an Alternative Unified Framework for Control of Non-regular Dynamics
G. Rega, S. Lenci, and J.M.T. Thompson
1 Controlling Chaos: A Hot Topic at the Change of the Millennium
2 The Paradigmatic OGY Method for Chaos Control
3 Use of OGY Method for Control of Chaos in Mechanics
3.1 The Pendulum System
3.2 Smooth Archetypal Oscillators
3.3 Vibro-Impact and Friction Systems
3.4 Coupled Mechanical Systems
3.5 Targeting in Astrodynamics
3.6 Atomic Force Microscopy
4 An Alternative Unified Framework for Control of Non-regular Dynamics of Mechanical Systems
4.1 Single Degree-of-Freedom Systems
4.2 Different Kinds of Global Bifurcations
4.3 Distance Between Stable and Unstable Manifolds
4.3.1 Effect of Damping
4.3.2 Effect of Excitation
4.3.3 The Perturbed Manifold Distance
4.3.4 More General Excitation and Damping.
4.3.5 Reference (Natural) and Controlling Excitations
4.3.6 Energetic Derivation of Perturbed Manifolds Distance
4.3.7 Minimum Manifolds Distance
4.4 Influence of the Parameters on the Manifolds Distance
4.5 Homoclinic Bifurcation Thresholds
4.6 Control Ideas
4.7 Gains and Saved Region
4.8 Optimal Control and Optimization Problems
4.8.1 Universal Optimization Problem
4.8.2 From the Universal Optimal Solution to the Real Optimal Excitation
4.9 Extended (Global'') and Localized (One-Side'') Application of Control
4.9.1 Global Control of Gains
4.9.2 Global Control of Homoclinic Bifurcation Thresholds
4.10 On the Application of the Control Methods to Archetypal Single-d.o.f. Systems
4.10.1 Smooth vs. Non-smooth Systems
4.10.2 Single-Well vs. Multi-Well Potentials
4.10.3 Softening vs. Hardening Systems
4.10.4 Homoclinic vs. Heteroclinic Bifurcations
4.10.5 Symmetric vs. Asymmetric Systems
4.10.6 Transient vs. Steady Dynamics
4.10.7 Overall vs. Localized Control
4.10.8 System-Independent vs. System-Dependent Controls
4.10.9 Finite- vs. Infinite-Dimensional Systems
References
Detection of Patterns Within Randomness
Ruedi Stoop and Markus Christen
1 Introduction and Overview
2 Log--log Steps in the Correlation Integral
3 Noiseless Single Patterns and Beyond
4 Analytical Derivation of s(n, m)
5 Main Theorem
6 Discussion and Outlook
References
Index