Handbook of Chaos Control

Heinz Georg Schuster is Professor of Theoretical Physics at the University of Kiel in Germany. In 1971 he received his doctorate and in 1976 he was appointed Professor at the University of Frankfurt/Main in Germany. He was a visiting professor at the Weizmann-Institute of Science in Israel and at the California Institute of Technology in Pasadena, USA. Professor Schuster works on the dynamical behaviour of complex adaptive systems and authored and coauthored several books in this field. His book "Deterministic Chaos" which was also published at Wiley-VCH, has been translated into five languages. Eckehard Scholl received his M.Sc. in physics from the University of Tuebingen, Germany, and his Ph.D. degree in applied mathematics from the University of Southampton, England. In 1989 he was appointed to a professorship in theoretical physics at the Technical University of Berlin, where he still teaches. His research interests are nonlinear dynamic systems, including nonlinear spatio-temporal dynamics, chaos, pattern formation, noise, and control. He authored and coauthored several books in his field.Professor Scholl was awarded the "Champion in teaching" prize by the Technical University of Berlin in 1997 and a Visiting Professorship by the London Mathematical Society in 2004.

"The book is interdisciplinary and can be of interest to graduate students, researchers in different fields: physicists, mathematicians and, engineers." (Zentralblatt MATH, 1132, 2008)

• Preface xxiList of Contributors xxiiiPart I Basic Aspects and Extension of Methods1 Controlling Chaos 3Elbert E. N. Macau and Celso Grebogi1.1 Introduction 31.2 The OGY Chaos Control 61.3 Targeting–Steering Chaotic Trajectories 81.3.1 Part I: Finding a Proper Trajectory 91.3.2 Part II: Finding a Pseudo-Orbit Trajectory 101.3.3 The Targeting Algorithm 121.4 Applying Control of Chaos and Targeting Ideas 131.4.1 Controlling an Electronic Circuit 131.4.2 Controlling a Complex System 191.5 Conclusion 26References 262 Time-Delay Control for Discrete Maps 29Joshua E. S. Socolar2.1 Overview: Why Study Discrete Maps? 292.2 Theme and Variations 312.2.1 Rudimentary Time-Delay Feedback 322.2.2 Extending the Domain of Control 342.2.3 High-Dimensional Systems 372.3 Robustness of Time-Delay Stabilization 412.4 Summary 44Acknowledgments 44References 443 An Analytical Treatment of the Delayed Feedback Control Algorithm 47Kestutis Pyragas, Tatjana Pyragienė, and Viktoras Pyragas3.1 Introduction 473.2 Proportional Versus Delayed Feedback 503.3 Controlling Periodic Orbits Arising from a Period Doubling Bifurcation 533.3.1 Example: Controlling the Rössler System 543.4 Control of Forced Self-Sustained Oscillations 573.4.1 Problem Formulation and Averaged Equation 573.4.2 Periodic Orbits of the Free System 583.4.3 Linear Stability of the System Controlled by Delayed Feedback 603.4.4 Numerical Demonstrations 633.5 Controlling Torsion-Free Periodic Orbits 633.5.1 Example: Controlling the Lorenz System at a Subcritical Hopf Bifurcation 653.6 Conclusions 68References 704 Beyond the Odd-Number Limitation of Time-Delayed Feedback Control 73Bernold Fiedler, Valentin Flunkert, Marc Georgi, Philipp Hövel, and Eckehard Schöll4.1 Introduction 734.2 Mechanism of Stabilization 744.3 Conditions on the Feedback Gain 784.4 Conclusion 82Acknowledgments 82Appendix: Calculation of Floquet Exponents 82References 835 On Global Properties of Time-Delayed Feedback Control 85Wolfram Just5.1 Introduction 855.2 A Comment on Control and Root Finding Algorithms 885.3 Codimension-Two Bifurcations and Basins of Attraction 915.3.1 The Transition from Super- to Subcritical Behavior 915.3.2 Probing Basins of Attraction in Experiments 935.4 A Case Study of Global Features for Time-Delayed Feedback Control 945.4.1 Analytical Bifurcation Analysis of One-Dimensional Maps 955.4.2 Dependence of Sub- and Supercritical Behavior on the Observable 985.4.3 Influence of the Coupling of the Control Force 995.5 Conclusion 101Acknowledgments 102Appendix A. Normal Form Reduction 103Appendix B. Super- and Subcritical Hopf Bifurcation for Maps 106References 1066 Poincaré-Based Control of Delayed Measured Systems: Limitations and Improved Control 109Jens Christian Claussen6.1 Introduction 1096.1.1 The Delay Problem–Time-Discrete Case 1096.1.2 Experimental Setups with Delay 1116.2 Ott-Grebogi-Yorke (OGY) Control 1126.3 Limitations of Unmodified Control and Simple Improved Control Schemes 1136.3.1 Limitations of Unmodified OGY Control in the Presence of Delay 1136.3.2 Stability Diagrams Derived by the Jury Criterion 1166.3.3 Stabilizing Unknown Fixed Points: Limitations of Unmodified Difference Control 1166.3.4 Rhythmic Control Schemes: Rhythmic OGY Control 1196.3.5 Rhythmic Difference Control 1206.3.6 A Simple Memory Control Scheme: Using State Space Memory 1226.4 Optimal Improved Control Schemes 1236.4.1 Linear Predictive Logging Control (LPLC) 1236.4.2 Nonlinear Predictive Logging Control 1246.4.3 Stabilization of Unknown Fixed Points: Memory Difference Control (mdc) 1256.5 Summary 126References 1277 Nonlinear and Adaptive Control of Chaos 129Alexander Fradkov and Alexander Pogromsky7.1 Introduction 1297.2 Chaos and Control: Preliminaries 1307.2.1 Definitions of Chaos 1307.2.2 Models of Controlled Systems 1317.2.3 Control Goals 1327.3 Methods of Nonlinear Control 1347.3.1 Gradient Method 1357.3.2 Speed-Gradient Method 1367.3.3 Feedback Linearization 1417.3.4 Other Methods 1427.3.5 Gradient Control of the Hénon System 1447.3.6 Feedback Linearization Control of the Lorenz System 1467.3.7 Speed-Gradient Stabilization of the Equilibrium Point for the Thermal Convection Loop Model 1477.4 Adaptive Control 1487.4.1 General Definitions 1487.4.2 Adaptive Master-Slave Synchronization of Rössler Systems 1497.5 Other Problems 1547.6 Conclusions 155Acknowledgment 155References 156Part II Controlling Space-time Chaos8 Localized Control of Spatiotemporal Chaos 161Roman O. Grigoriev and Andreas Handel8.1 Introduction 1618.1.1 Empirical Control 1638.1.2 Model-Based Control 1648.2 Symmetry and the Minimal Number of Sensors/Actuators 1678.3 Nonnormality and Noise Amplification 1708.4 Nonlinearity and the Critical Noise Level 1758.5 Conclusions 177References 1779 Controlling Spatiotemporal Chaos: The Paradigm of the Complex Ginzburg-Landau Equation 181Stefano Boccaletti and Jean Bragard9.1 Introduction 1819.2 The Complex Ginzburg-Landau Equation 1839.2.1 Dynamics Characterization 1859.3 Control of the CGLE 1879.4 Conclusions and Perspectives 192Acknowledgment 193References 19310 Multiple Delay Feedback Control 197Alexander Ahlborn and Ulrich Parlitz10.1 Introduction 19710.2 Multiple Delay Feedback Control 19810.2.1 Linear Stability Analysis 19910.2.2 Example: Colpitts Oscillator 20010.2.3 Comparison with High-Pass Filter and PD Controller 20310.2.4 Transfer Function of MDFC 20410.3 From Multiple Delay Feedback Control to Notch Filter Feedback 20610.4 Controllability Criteria 20810.4.1 Multiple Delay Feedback Control 20910.4.2 Notch Filter Feedback and High-Pass Filter 21010.5 Laser Stabilization Using MDFC and NFF 21110.6 Controlling Spatiotemporal Chaos 21310.6.1 The Ginzburg-Landau Equation 21310.6.2 Controlling Traveling Plane Waves 21410.6.3 Local Feedback Control 21510.7 Conclusion 218References 219Part III Controlling Noisy Motion11 Control of Noise-Induced Dynamics 223Natalia B. Janson, Alexander G. Balanov, and Eckehard Schöll11.1 Introduction 22311.2 Noise-Induced Oscillations Below Andronov-Hopf Bifurcation and their Control 22611.2.1 Weak Noise and Control: Correlation Function 22811.2.2 Weak Noise and No Control: Correlation Time and Spectrum 22911.2.3 Weak Noise and Control: Correlation Time 23111.2.4 Weak Noise and Control: Spectrum 23511.2.5 Any Noise and No Control: Correlation Time 23611.2.6 Any Noise and Control: Correlation Time and Spectrum 23811.2.7 So, What Can We Control? 24011.3 Noise-Induced Oscillations in an Excitable System and their Control 24111.3.1 Coherence Resonance in the FitzHugh-Nagumo System 24311.3.2 Correlation Time and Spectrum when Feedback is Applied 24411.3.3 Control of Synchronization in Coupled FitzHugh-Nagumo Systems 24511.3.4 What can We Control in an Excitable System? 24611.4 Delayed Feedback Control of Noise-Induced Pulses in a Model of an Excitable Medium 24711.4.1 Model Description 24711.4.2 Characteristics of Noise-Induced Patterns 24911.4.3 Control of Noise-Induced Patterns 25111.4.4 Mechanisms of Delayed Feedback Control of the Excitable Medium 25311.4.5 What Can Be Controlled in an Excitable Medium? 25411.5 Delayed Feedback Control of Noise-Induced Patterns in a Globally Coupled Reaction–Diffusion Model 25511.5.1 Spatiotemporal Dynamics in the Uncontrolled Deterministic System 25611.5.2 Noise-Induced Patterns in the Uncontrolled System 25811.5.3 Time-Delayed Feedback Control of Noise-Induced Patterns 26011.5.4 Linear Modes of the Inhomogeneous Fixed Point 26411.5.5 Delay-Induced Oscillatory Patterns 26811.5.6 What Can Be Controlled in a Globally Coupled Reaction–Diffusion System? 26911.6 Summary and Conclusions 270Acknowledgments 270References 27012 Controlling Coherence of Noisy and Chaotic Oscillators by Delayed Feedback 275Denis Goldobin, Michael Rosenblum, and Arkady Pikovsky12.1 Control of Coherence: Numerical Results 27612.1.1 Noisy Oscillator 27612.1.2 Chaotic Oscillator 27712.1.3 Enhancing Phase Synchronization 27912.2 Theory of Coherence Control 27912.2.1 Basic Phase Model 27912.2.2 Noise-Free Case 28012.2.3 Gaussian Approximation 28012.2.4 Self-Consistent Equation for Diffusion Constant 28212.2.5 Comparison of Theory and Numerics 28312.3 Control of Coherence by Multiple Delayed Feedback 28312.4 Conclusion 288References 28913 Resonances Induced by the Delay Time in Nonlinear Autonomous Oscillators with Feedback 291Cristina MasollerAcknowledgment 298References 299Part IV Communicating with Chaos, Chaos Synchronization14 Secure Communication with Chaos Synchronization 303Wolfgang Kinzel and Ido Kanter14.1 Introduction 30314.2 Synchronization of Chaotic Systems 30414.3 Coding and Decoding Secret Messages in Chaotic Signals 30914.4 Analysis of the Exchanged Signal 31114.5 Neural Cryptography 31314.6 Public Key Exchange by Mutual Synchronization 31514.7 Public Keys by Asymmetric Attractors 31814.8 Mutual Chaos Pass Filter 31914.9 Discussion 321References 32315 Noise Robust Chaotic Systems 325Thomas L. Carroll15.1 Introduction 32515.2 Chaotic Synchronization 32615.3 2-Frequency Self-Synchronizing Chaotic Systems 32615.3.1 Simple Maps 32615.4 2-Frequency Synchronization in Flows 32915.4.1 2-Frequency Additive Rössler 32915.4.2 Parameter Variation and Periodic Orbits 33215.4.3 Unstable Periodic Orbits 33315.4.4 Floquet Multipliers 33415.4.5 Linewidths 33515.5 Circuit Experiments 33615.5.1 Noise Effects 33815.6 Communication Simulations 33815.7 Multiplicative Two-Frequency Rössler Circuit 34115.8 Conclusions 346References 34616 Nonlinear Communication Strategies 349Henry D.I. Abarbanel16.1 Introduction 34916.1.1 Secrecy, Encryption, and Security? 35016.2 Synchronization 35116.3 Communicating Using Chaotic Carriers 35316.4 Two Examples from Optical Communication 35516.4.1 Rare-Earth-Doped Fiber Amplifier Laser 35516.4.2 Time Delay Optoelectronic Feedback Semiconductor Laser 35716.5 Chaotic Pulse Position Communication 35916.6 Why Use Chaotic Signals at All? 36216.7 Undistorting the Nonlinear Effects of the Communication Channel 36316.8 Conclusions 366References 36717 Synchronization and Message Transmission for Networked Chaotic Optical Communications 369K. Alan Shore, Paul S. Spencer, and Ilestyn Pierce17.1 Introduction 36917.2 Synchronization and Message Transmission 37017.3 Networked Chaotic Optical Communication 37217.3.1 Chaos Multiplexing 37317.3.2 Message Relay 37317.3.3 Message Broadcasting 37417.4 Summary 376Acknowledgments 376References 37618 Feedback Control Principles for Phase Synchronization 379Vladimir N. Belykh, Grigory V. Osipov, and Jürgen Kurths18.1 Introduction 37918.2 General Principles of Automatic Synchronization 38118.3 Two Coupled Poincaré Systems 38418.4 Coupled van der Pol and Rössler Oscillators 38618.5 Two Coupled Rössler Oscillators 38918.6 Coupled Rössler and Lorenz Oscillators 39118.7 Principles of Automatic Synchronization in Networks of Coupled Oscillators 39318.8 Synchronization of Locally Coupled Regular Oscillators 39518.9 Synchronization of Locally Coupled Chaotic Oscillators 39718.10 Synchronization of Globally Coupled Chaotic Oscillators 39918.11 Conclusions 401References 401Part V Applications to Optics19 Controlling Fast Chaos in Optoelectronic Delay Dynamical Systems 407Lucas Illing, Daniel J. Gauthier, and Jonathan N. Blakely19.1 Introduction 40719.2 Control-Loop Latency: A Simple Example 40819.3 Controlling Fast Systems 41219.4 A Fast Optoelectronic Chaos Generator 41519.5 Controlling the Fast Optoelectronic Device 41919.6 Outlook 423Acknowledgment 424References 42420 Control of Broad-Area Laser Dynamics with Delayed Optical Feedback 427Nicoleta Gaciu, Edeltraud Gehrig, and Ortwin Hess20.1 Introduction: Spatiotemporally Chaotic Semiconductor Lasers 42720.2 Theory: Two-Level Maxwell-Bloch Equations 42920.3 Dynamics of the Solitary Laser 43220.4 Detection of Spatiotemporal Complexity 43320.4.1 Reduction of the Number of Modes by Coherent Injection 43320.4.2 Pulse-Induced Mode Synchronization 43520.5 Self-Induced Stabilization and Control with Delayed Optical Feedback 43820.5.1 Influence of Delayed Optical Feedback 43920.5.2 Influence of the Delay Time 44020.5.3 Spatially Structured Delayed Optical Feedback Control 44420.5.4 Filtered Spatially Structured Delayed Optical Feedback 44920.6 Conclusions 451References 45321 Noninvasive Control of Semiconductor Lasers by Delayed Optical Feedback 455Hans-Jürgen Wünsche, Sylvia Schikora, and Fritz Henneberger21.1 The Role of the Optical Phase 45621.2 Generic Linear Model 45921.3 Generalized Lang-Kobayashi Model 46121.4 Experiment 46221.4.1 The Integrated Tandem Laser 46321.4.2 Design of the Control Cavity 46421.4.3 Maintaining Resonance 46521.4.4 Latency and Coupling Strength 46521.4.5 Results of the Control Experiment 46621.5 Numerical Simulation 46821.5.1 Traveling-Wave Model 46821.5.2 Noninvasive Control Beyond a Hopf Bifurcation 47021.5.3 Control Dynamics 47021.5.4 Variation of the Control Parameters 47121.6 Conclusions 473Acknowledgment 473References 47322 Chaos and Control in Semiconductor Lasers 475Junji Ohtsubo22.1 Introduction 47522.2 Chaos in Semiconductor Lasers 47622.2.1 Laser Chaos 47622.2.2 Optical Feedback Effects in Semiconductor Lasers 47822.2.3 Chaotic Effects in Newly Developed Semiconductor Lasers 48022.3 Chaos Control in Semiconductor Lasers 48522.4 Control in Newly Developed Semiconductor Lasers 49422.5 Conclusions 497References 49823 From Pattern Control to Synchronization: Control Techniques in Nonlinear Optical Feedback Systems 501Björn Gütlich and Cornelia Denz23.1 Control Methods for Spatiotemporal Systems 50223.2 Optical Single-Feedback Systems 50323.2.1 A Simplified Single-Feedback Model System 50423.2.2 The Photorefractive Single-Feedback System – Coherent Nonlinearity 50623.2.3 Theoretical Description of the Photorefractive Single-Feedback System 50823.2.4 Linear Stability Analysis 50923.2.5 The LCLV Single-Feedback System – Incoherent Nonlinearity 51023.2.6 Phase-Only Mode 51123.2.7 Polarization Mode 51323.2.8 Dissipative Solitons in the LCLV Feedback System 51323.3 Spatial Fourier Control 51423.3.1 Experimental Determination of Marginal Instability 51623.3.2 Stabilization of Unstable Pattern 51723.3.3 Direct Fourier Filtering 51823.3.4 Positive Fourier Control 51823.3.5 Noninvasive Fourier Control 51923.4 Real-Space Control 52023.4.1 Invasive Forcing 52023.4.2 Positioning of Localized States 52223.4.3 System Homogenization 52223.4.4 Static Positioning 52323.4.5 Addressing and Dynamic Positioning 52323.5 Spatiotemporal Synchronization 52423.5.1 Spatial Synchronization of Periodic Pattern 52423.5.2 Unidirectional Synchronization of Two LCLV Systems 52523.5.3 Synchronization of Spatiotemporal Complexity 52623.6 Conclusions and Outlook 527References 528Part VI Applications to Electronic Systems24 Delayed-Feedback Control of Chaotic Spatiotemporal Patterns in Semiconductor Nanostructures 533Eckehard Schöll24.1 Introduction 53324.2 Control of Chaotic Domain and Front Patterns in Superlattices 53624.3 Control of Chaotic Spatiotemporal Oscillations in Resonant Tunneling Diodes 54424.4 Conclusions 553Acknowledgments 554References 55425 Observing Global Properties of Time-Delayed Feedback Control in Electronic Circuits 559Hartmut Benner, Chol-Ung Choe, Klaus Höhne, Clemens von Loewenich, Hiroyuki Shirahama, and Wolfram Just25.1 Introduction 55925.2 Discontinuous Transitions for Extended Time-Delayed Feedback Control 56025.2.1 Theoretical Considerations 56025.2.2 Experimental Setup 56125.2.3 Observation of Bistability 56225.2.4 Basin of Attraction 56425.3 Controlling Torsion-Free Unstable Orbits 56525.3.1 Applying the Concept of an Unstable Controller 56725.3.2 Experimental Design of an Unstable van der Pol Oscillator 56725.3.3 Control Coupling and Basin of Attraction 56925.4 Conclusions 572References 57326 Application of a Black Box Strategy to Control Chaos 575Achim Kittel and Martin Popp26.1 Introduction 57526.2 The Model Systems 57526.2.1 Shinriki Oscillator 57626.2.2 Mackey-Glass Type Oscillator 57726.3 The Controller 58026.4 Results of the Application of the Controller to the Shinriki Oscillator 58226.4.1 Spectroscopy of Unstable Periodic Orbits 58426.5 Results of the Application of the Controller to the Mackey-Glass Oscillator 58526.5.1 Spectroscopy of Unstable Periodic Orbits 58726.6 Further Improvements 58926.7 Conclusions 589Acknowledgment 590References 590Part VII Applications to Chemical Reaction Systems27 Feedback-Mediated Control of Hypermeandering Spiral Waves 593Jan Schlesner, Vladimir Zykov, and Harald Engel27.1 Introduction 59327.2 The FitzHugh-Nagumo Model 59427.3 Stabilization of Rigidly Rotating Spirals in the Hypermeandering Regime 59627.4 Control of Spiral Wave Location in the Hypermeandering Regime 59927.5 Discussion 605References 60628 Control of Spatiotemporal Chaos in Surface Chemical Reactions 609Carsten Beta and Alexander S. Mikhailov28.1 Introduction 60928.2 The Catalytic CO Oxidation on Pt(110) 61028.2.1 Mechanism 61028.2.2 Modeling 61128.2.3 Experimental Setup 61228.3 Spatiotemporal Chaos in Catalytic CO Oxidation on Pt(110) 61328.4 Control of Spatiotemporal Chaos by Global Delayed Feedback 61528.4.1 Control of Turbulence in Catalytic CO Oxidation – Experimental 61628.4.1.1 Control of Turbulence 61728.4.1.2 Spatiotemporal Pattern Formation 61828.4.2 Control of Turbulence in Catalytic CO Oxidation – Numerical Simulations 61928.4.3 Control of Turbulence in Oscillatory Media – Theory 62128.4.4 Time-Delay Autosynchronization 62528.5 Control of Spatiotemporal Chaos by Periodic Forcing 628Acknowledgment 630References 63029 Forcing and Feedback Control of Arrays of Chaotic Electrochemical Oscillators 633István Z. Kiss and John L. Hudson29.1 Introduction 63329.2 Control of Single Chaotic Oscillator 63429.2.1 Experimental Setup 63429.2.2 Chaotic Ni Dissolution: Low-Dimensional, Phase Coherent Attractor 63529.2.2.1 Unforced Chaotic Oscillator 63529.2.2.2 Phase of the Unforced System 63629.2.3 Forcing: Phase Synchronization and Intermittency 63729.2.3.1 Forcing with X=x 0 63729.2.3.2 Forcing with X 6ˆ X 0 63829.2.4 Delayed Feedback: Tracking 63829.3 Control of Small Assemblies of Chaotic Oscillators 64029.4 Control of Oscillator Populations 64229.4.1 Global Coupling 64229.4.2 Periodic Forcing of Arrays of Chaotic Oscillators 64329.4.3 Feedback on Arrays of Chaotic Oscillators 64429.4.4 Feedback, Forcing, and Global Coupling: Order Parameter 64529.4.5 Control of Complexity of a Collective Signal 64629.5 Concluding Remarks 647Acknowledgment 648References 649Part VIII Applications to Biology30 Control of Synchronization in Oscillatory Neural Networks 653Peter A. Tass, Christian Hauptmann, and Oleksandr V. Popovych30.1 Introduction 65330.2 Multisite Coordinated Reset Stimulation 65430.3 Linear Multisite Delayed Feedback 66230.4 Nonlinear Delayed Feedback 66630.5 Reshaping Neural Networks 67430.6 Discussion 676References 67831 Control of Cardiac Electrical Nonlinear Dynamics 683Trine Krogh-Madsen, Peter N. Jordan, and David J. Christini31.1 Introduction 68331.2 Cardiac Electrophysiology 68431.2.1 Restitution and Alternans 68531.3 Cardiac Arrhythmias 68631.3.1 Reentry 68731.3.2 Ventricular Tachyarrhythmias 68831.3.3 Alternans as an Arrhythmia Trigger 68831.4 Current Treatment of Arrhythmias 68931.4.1 Pharmacological Treatment 68931.4.2 Implantable Cardioverter Defibrillators 68931.4.3 Ablation Therapy 69031.5 Alternans Control 69131.5.1 Controlling Cellular Alternans 69131.5.2 Control of Alternans in Tissue 69231.5.3 Limitations of the DFC Algorithm in Alternans Control 69331.5.4 Adaptive DI Control 69431.6 Control of Ventricular Tachyarrhythmias 69531.6.1 Suppression of Spiral Waves 69631.6.2 Antitachycardia Pacing 69631.6.3 Unpinning Spiral Waves 69831.7 Conclusions and Prospects 699References 70032 Controlling Spatiotemporal Chaos and Spiral Turbulence in Excitable Media 703Sitabhra Sinha and S. Sridhar32.1 Introduction 70332.2 Models of Spatiotemporal Chaos in Excitable Media 70632.3 Global Control 70832.4 Nonglobal Spatially Extended Control 71132.4.1 Applying Control Over a Mesh 71132.4.2 Applying Control Over an Array of Points 71332.5 Local Control of Spatiotemporal Chaos 71432.6 Discussion 716Acknowledgments 717References 718Part IX Applications to Engineering33 Nonlinear Chaos Control and Synchronization 721Henri J. C. Huijberts and Henk Nijmeijer33.1 Introduction 72133.2 Nonlinear Geometric Control 72133.2.1 Some Differential Geometric Concepts 72233.2.2 Nonlinear Controllability 72333.2.3 Chaos Control Through Feedback Linearization 72833.2.4 Chaos Control Through Input–Output Linearization 73233.3 Lyapunov Design 73733.3.1 Lyapunov Stability and Lyapunov’s First Method 73733.3.2 Lyapunov’s Direct Method 73933.3.3 LaSalle’s Invariance Principle 74133.3.4 Examples 742References 74934 Electronic Chaos Controllers – From Theory to Applications 751Maciej Ogorzałek34.1 Introduction 75134.1.1 Chaos Control 75234.1.2 Fundamental Properties of Chaotic Systems and Goals of the Control 75334.2 Requirements for Electronic Implementation of Chaos Controllers 75434.3 Short Description of the OGY Technique 75534.4 Implementation Problems for the OGY Method 75734.4.1 Effects of Calculation Precision 75834.4.2 Approximate Procedures for Finding Periodic Orbits 75934.4.3 Effects of Time Delays 75934.5 Occasional Proportional Feedback (Hunt’s) Controller 76134.5.1 Improved Chaos Controller for Autonomous Circuits 76334.6 Experimental Chaos Control Systems 76534.6.1 Control of a Magnetoelastic Ribbon 76534.6.2 Control of a Chaotic Laser 76634.6.3 Chaos-Based Arrhythmia Suppression and Defibrillation 76734.7 Conclusions 768References 76935 Chaos in Pulse-Width Modulated Control Systems 771Zhanybai T. Zhusubaliyev and Erik Mosekilde35.1 Introduction 77135.2 DC/DC Converter with Pulse-Width Modulated Control 77435.3 Bifurcation Analysis for the DC/DC Converter with One-Level Control 77835.4 DC/DC Converter with Two-Level Control 78135.5 Bifurcation Analysis for the DC/DC Converter with Two-Level Control 78335.6 Conclusions 784Acknowledgments 788References 78836 Transient Dynamics of Duffing System Under Time-Delayed Feedback Control: Global Phase Structure and Application to Engineering 793Takashi Hikihara and Kohei Yamasue36.1 Introduction 79336.2 Transient Dynamics of Transient Behavior 79436.2.1 Magnetoelastic Beam and Experimental Setup 79436.2.2 Transient Behavior 79536.3 Initial Function and Domain of Attraction 79736.4 Persistence of Chaos 80036.5 Application of TDFC to Nanoengineering 80336.5.1 Dynamic Force Microscopy and its Dynamics 80336.5.2 Application of TDFC 80536.5.3 Extension of Operating Range 80636.6 Conclusions 808References 808Subject Index 811