Schoell E., Schuster H.G. (eds.) Handbook of Chaos Control

N.-Y.: Wiley-VCH, 2007. - 852p.This long-awaited revised second edition of the standard reference on the subject has been considerably expanded to include such recent developments as novel control schemes, control of chaotic space-time patterns, control of noisy nonlinear systems, and communication with chaos, as well as promising new directions in research. The contributions from leading international scientists active in the field provide a comprehensive overview of our current level of knowledge on chaos control and its applications in physics, chemistry, biology, medicine, and engineering. In addition, they show the overlap with the traditional field of control theory in the engineering community.An interdisciplinary approach of interest to scientists and engineers working in a number of areas.List of ContributorsBasic Aspects and Extension of MethodsThe OGY Chaos Control
Targeting–Steering Chaotic Trajectories
Part I: Finding a Proper Trajectory
Part II: Finding a Pseudo-Orbit Trajectory
The Targeting Algorithm
Controlling an Electronic Circuit
Controlling a Complex SystemOverview: Why Study Discrete Maps?
Theme and Variations
Rudimentary Time-Delay Feedback
Extending the Domain of Control
High-Dimensional Systems
Robustness of Time-Delay StabilizationProportional Versus Delayed Feedback
Controlling Periodic Orbits Arising from a Period Doubling Bifurcation
Example: Controlling the Rössler System
Problem Formulation and Averaged Equation
Periodic Orbits of the Free System
Linear Stability of the System Controlled by Delayed Feedback
Controlling Torsion-Free Periodic Orbits
Example: Controlling the Lorenz System at a Subcritical Hopf Bifurcation
ConclusionsMechanism of Stabilization
Conditions on the Feedback Gain
Appendix: Calculation of Floquet ExponentsA Comment on Control and Root Finding Algorithms
The Transition from Super- to Subcritical Behavior
Probing Basins of Attraction in Experiments
A Case Study of Global Features for Time-Delayed Feedback Control
Analytical Bifurcation Analysis of One-Dimensional Maps
Dependence of Sub- and Supercritical Behavior on the Observable
Influence of the Coupling of the Control ForceAppendix A Normal Form ReductionThe Delay Problem–Time-Discrete Case
Experimental Setups with Delay
Ott-Grebogi-Yorke (OGY) Control
Limitations of Unmodified OGY Control in the Presence of Delay
Stabilizing Unknown Fixed Points: Limitations of Unmodified Difference Control
Rhythmic Control Schemes: Rhythmic OGY Control
Rhythmic Difference Control
A Simple Memory Control Scheme: Using State Space Memory
Linear Predictive Logging Control (LPLC)
Nonlinear Predictive Logging Control
Stabilization of Unknown Fixed Points: Memory Difference Control (MDC)Definitions of Chaos
Models of Controlled Systems
Control Goals
Methods of Nonlinear Control
Gradient Method
Speed-Gradient Method
Feedback Linearization
Other Methods
Gradient Control of the Hénon System
Feedback Linearization Control of the Lorenz System
Speed-Gradient Stabilization of the Equilibrium Point for the Thermal Convection Loop Model
General Definitions
Adaptive Master-Slave Synchronization of Rössler Systems
Other ProblemsControlling Space-Time ChaosEmpirical Control
Model-Based Control
Symmetry and the Minimal Number of Sensors/Actuators
Nonnormality and Noise Amplification
Nonlinearity and the Critical Noise LevelThe Complex Ginzburg-Landau Equation
Dynamics Characterization
Control of the CGLE
Conclusions and PerspectivesMultiple Delay Feedback Control
Linear Stability Analysis
Example: Colpitts Oscillator
Comparison with High-Pass Filter and PD Controller
Transfer Function of MDFC
From Multiple Delay Feedback Control to Notch Filter Feedback
Controllability Criteria
Multiple Delay Feedback Control
Notch Filter Feedback and High-Pass Filter
Laser Stabilization Using MDFC and NFF
The Ginzburg-Landau Equation
Controlling Traveling Plane Waves
Local Feedback ControlControlling Noisy MotionNoise-Induced Oscillations Below Andronov-Hopf Bifurcation and their Control
Weak Noise and Control: Correlation Function
Weak Noise and No Control: Correlation Time and Spectrum
Weak Noise and Control: Correlation Time
Weak Noise and Control: Spectrum
Any Noise and No Control: Correlation Time
Any Noise and Control: Correlation Time and Spectrum
So, What Can We Control?
Noise-Induced Oscillations in an Excitable System and their Control
Coherence Resonance in the FitzHugh-Nagumo System
Correlation Time and Spectrum when Feedback is Applied
Control of Synchronization in Coupled FitzHugh-Nagumo Systems
What can We Control in an Excitable System?
Model Description
Characteristics of Noise-Induced Patterns
Control of Noise-Induced Patterns
Mechanisms of Delayed Feedback Control of the Excitable Medium
What Can Be Controlled in an Excitable Medium?
Delayed Feedback Control of Noise-Induced Patterns in a Globally Coupled Reaction–Diffusion Model
Spatiotemporal Dynamics in the Uncontrolled Deterministic System
Noise-Induced Patterns in the Uncontrolled System
Time-Delayed Feedback Control of Noise-Induced Patterns
Linear Modes of the Inhomogeneous Fixed Point
Delay-Induced Oscillatory Patterns
What Can Be Controlled in a Globally Coupled Reaction–Diffusion System?Controlling Coherence of Noisy and Chaotic Oscillators by Delayed Feedback
Noisy Oscillator
Chaotic Oscillator
Basic Phase Model
Gaussian Approximation
Self-Consistent Equation for Diffusion Constant
Control of Coherence by Multiple Delayed FeedbackResonances Induced by the Delay Time in Nonlinear Autonomous Oscillators with FeedbackCommunicating with Chaos, Chaos SynchronizationSynchronization of Chaotic Systems
Coding and Decoding Secret Messages in Chaotic Signals
Analysis of the Exchanged Signal
Neural Cryptography
Public Key Exchange by Mutual Synchronization
Public Keys by Asymmetric Attractors
Mutual Chaos Pass Filter
DiscussionSimple Maps
-Frequency Additive Rössler
Parameter Variation and Periodic Orbits
Unstable Periodic Orbits
Floquet Multipliers
Linewidths
Circuit Experiments
Communication Simulations
Multiplicative Two-Frequency Rössler CircuitSecrecy, Encryption, and Security?
Synchronization
Communicating Using Chaotic Carriers
Rare-Earth-Doped Fiber Amplifier Laser
Time Delay Optoelectronic Feedback Semiconductor Laser
Chaotic Pulse Position Communication
Why Use Chaotic Signals at All?
Undistorting the Nonlinear Effects of the Communication Channel
ConclusionsSynchronization and Message Transmission
Networked Chaotic Optical Communication
Message Relay
Message BroadcastingGeneral Principles of Automatic Synchronization
Two Coupled Poincaré Systems
Coupled van der Pol and Rössler Oscillators
Two Coupled Rössler Oscillators
Coupled Rössler and Lorenz Oscillators
Principles of Automatic Synchronization in Networks of Coupled Oscillators
Synchronization of Locally Coupled Regular Oscillators
Synchronization of Locally Coupled Chaotic Oscillators
Synchronization of Globally Coupled Chaotic OscillatorsApplications to OpticsControl-Loop Latency: A Simple Example
Controlling Fast Systems
A Fast Optoelectronic Chaos Generator
Controlling the Fast Optoelectronic Device
OutlookIntroduction: Spatiotemporally Chaotic Semiconductor Lasers
Theory: Two-Level Maxwell-Bloch Equations
Dynamics of the Solitary Laser
Reduction of the Number of Modes by Coherent Injection
Pulse-Induced Mode Synchronization
Self-Induced Stabilization and Control with Delayed Optical Feedback
Influence of Delayed Optical Feedback
Influence of the Delay Time
Spatially Structured Delayed Optical Feedback Control
Filtered Spatially Structured Delayed Optical Feedback
ConclusionsNoninvasive Control of Semiconductor Lasers by Delayed Optical Feedback
The Role of the Optical Phase
Generic Linear Model
Generalized Lang-Kobayashi Model
Experiment
The Integrated Tandem Laser
Design of the Control Cavity
Latency and Coupling Strength
Results of the Control Experiment
Traveling-Wave Model
Control Dynamics
Variation of the Control ParametersLaser Chaos
Optical Feedback Effects in Semiconductor Lasers
Chaotic Effects in Newly Developed Semiconductor Lasers
Chaos Control in Semiconductor Lasers
Control in Newly Developed Semiconductor Lasers
ConclusionsFrom Pattern Control to Synchronization: Control Techniques in Nonlinear Optical Feedback Systems
Control Methods for Spatiotemporal Systems
Optical Single-Feedback Systems
A Simplified Single-Feedback Model System
The Photorefractive Single-Feedback System – Coherent Nonlinearity
Theoretical Description of the Photorefractive Single-Feedback System
Linear Stability Analysis
The LCLV Single-Feedback System – Incoherent Nonlinearity
Phase-Only Mode
Dissipative Solitons in the LCLV Feedback System
Spatial Fourier Control
Experimental Determination of Marginal Instability
Stabilization of Unstable Pattern
Positive Fourier Control
Noninvasive Fourier Control
Invasive Forcing
System Homogenization
Addressing and Dynamic Positioning
Spatial Synchronization of Periodic Pattern
Unidirectional Synchronization of Two LCLV Systems
Synchronization of Spatiotemporal Complexity
Conclusions and OutlookApplications to Electronic SystemsControl of Chaotic Domain and Front Patterns in Superlattices
Control of Chaotic Spatiotemporal Oscillations in Resonant Tunneling Diodes
ConclusionsTheoretical Considerations
Experimental Setup
Observation of Bistability
Basin of Attraction
Controlling Torsion-Free Unstable Orbits
Experimental Design of an Unstable van der Pol Oscillator
Control Coupling and Basin of Attraction
ConclusionsThe Model Systems
Shinriki Oscillator
Mackey-Glass Type Oscillator
The Controller
Results of the Application of the Controller to the Shinriki Oscillator
Spectroscopy of Unstable Periodic Orbits
Results of the Application of the Controller to the Mackey-Glass Oscillator
Spectroscopy of Unstable Periodic Orbits
ConclusionsApplications to Chemical Reaction SystemsThe FitzHugh-Nagumo Model
Stabilization of Rigidly Rotating Spirals in the Hypermeandering Regime
Control of Spiral Wave Location in the Hypermeandering Regime
DiscussionMechanism
Modeling
Experimental Setup
Spatiotemporal Chaos in Catalytic CO Oxidation on Pt()
Control of Spatiotemporal Chaos by Global Delayed Feedback
Control of Turbulence in Catalytic CO Oxidation – Experimental
Control of Turbulence
Spatiotemporal Pattern Formation
Control of Turbulence in Catalytic CO Oxidation – Numerical Simulations
Control of Turbulence in Oscillatory Media – Theory
Time Delay Autosynchronization
Control of Spatiotemporal Chaos by Periodic ForcingExperimental Setup
Unforced Chaotic Oscillator
Phase of the Unforced System
Forcing with Ω = ω()
Delayed Feedback: Tracking
Control of Small Assemblies of Chaotic Oscillators
Global Coupling
Periodic Forcing of Arrays of Chaotic Oscillators
Feedback on Arrays of Chaotic Oscillators
Feedback, Forcing, and Global Coupling: Order Parameter
Control of Complexity of a Collective Signal
Concluding RemarksApplications to BiologyMultisite Coordinated Reset Stimulation
Linear Multisite Delayed Feedback
Nonlinear Delayed Feedback
Reshaping Neural Networks
DiscussionCardiac Electrophysiology
Restitution and Alternans
Cardiac Arrhythmias
Reentry
Alternans as an Arrhythmia Trigger
Implantable Cardioverter Defibrillators
Ablation Therapy
Controlling Cellular Alternans
Control of Alternans in Tissue
Limitations of the DFC Algorithm in Alternans Control
Adaptive DI Control
Control of Ventricular Tachyarrhythmias
Antitachycardia Pacing
Unpinning Spiral Waves
Conclusions and ProspectsModels of Spatiotemporal Chaos in Excitable Media
Global Control
Applying Control Over a Mesh
Applying Control Over an Array of Points
Local Control of Spatiotemporal Chaos
DiscussionApplications to EngineeringNonlinear Geometric Control
Some Differential Geometric Concepts
Nonlinear Controllability
Chaos Control Through Feedback Linearization
Chaos Control Through Input–Output Linearization
Lyapunov Stability and Lyapunov’s First Method
Lyapunov’s Direct Method
LaSalle’s Invariance Principle
ExamplesChaos Control
Fundamental Properties of Chaotic Systems and Goals of the Control
Requirements for Electronic Implementation of Chaos Controllers
Short Description of the OGY Technique
Implementation Problems for the OGY Method
Effects of Calculation Precision
Effects of Time Delays
Occasional Proportional Feedback (Hunt's) Controller
Improved Chaos Controller for Autonomous Circuits
Control of a Magnetoelastic Ribbon
Control of a Chaotic Laser
Chaos-Based Arrhythmia Suppression and Defibrillation
ConclusionsDC/DC Converter with Pulse-Width Modulated Control
Bifurcation Analysis for the DC/DC Converter with One-Level Control
DC/DC Converter with Two-Level Control
Bifurcation Analysis for the DC/DC Converter with Two-Level Control
ConclusionsMagnetoelastic Beam and Experimental Setup
Transient Behavior
Initial Function and Domain of Attraction
Persistence of Chaos
Dynamic Force Microscopy and its Dynamics
Application of TDFC
Extension of Operating RangeSubject Index