Eckehard Schöll & Heinz Georg Schuster 
Handbook of Chaos Control [PDF ebook] 

Preface xxi

List of Contributors xxiii

Part I Basic Aspects and Extension of Methods

1 Controlling Chaos 3
Elbert E. N. Macau and Celso Grebogi

1.1 Introduction 3

1.2 The OGY Chaos Control 6

1.3 Targeting–Steering Chaotic Trajectories 8

1.3.1 Part I: Finding a Proper Trajectory 9

1.3.2 Part II: Finding a Pseudo-Orbit Trajectory 10

1.3.3 The Targeting Algorithm 12

1.4 Applying Control of Chaos and Targeting Ideas 13

1.4.1 Controlling an Electronic Circuit 13

1.4.2 Controlling a Complex System 19

1.5 Conclusion 26

References 26

2 Time-Delay Control for Discrete Maps 29
Joshua E. S. Socolar

2.1 Overview: Why Study Discrete Maps? 29

2.2 Theme and Variations 31

2.2.1 Rudimentary Time-Delay Feedback 32

2.2.2 Extending the Domain of Control 34

2.2.3 High-Dimensional Systems 37

2.3 Robustness of Time-Delay Stabilization 41

2.4 Summary 44

Acknowledgments 44

References 44

3 An Analytical Treatment of the Delayed Feedback Control Algorithm 47
Kestutis Pyragas, Tatjana Pyragienė, and Viktoras Pyragas

3.1 Introduction 47

3.2 Proportional Versus Delayed Feedback 50

3.3 Controlling Periodic Orbits Arising from a Period Doubling Bifurcation 53

3.3.1 Example: Controlling the Rössler System 54

3.4 Control of Forced Self-Sustained Oscillations 57

3.4.1 Problem Formulation and Averaged Equation 57

3.4.2 Periodic Orbits of the Free System 58

3.4.3 Linear Stability of the System Controlled by Delayed Feedback 60

3.4.4 Numerical Demonstrations 63

3.5 Controlling Torsion-Free Periodic Orbits 63

3.5.1 Example: Controlling the Lorenz System at a Subcritical Hopf Bifurcation 65

3.6 Conclusions 68

References 70

4 Beyond the Odd-Number Limitation of Time-Delayed Feedback Control 73
Bernold Fiedler, Valentin Flunkert, Marc Georgi, Philipp Hövel, and Eckehard Schöll

4.1 Introduction 73

4.2 Mechanism of Stabilization 74

4.3 Conditions on the Feedback Gain 78

4.4 Conclusion 82

Acknowledgments 82

Appendix: Calculation of Floquet Exponents 82

References 83

5 On Global Properties of Time-Delayed Feedback Control 85
Wolfram Just

5.1 Introduction 85

5.2 A Comment on Control and Root Finding Algorithms 88

5.3 Codimension-Two Bifurcations and Basins of Attraction 91

5.3.1 The Transition from Super- to Subcritical Behavior 91

5.3.2 Probing Basins of Attraction in Experiments 93

5.4 A Case Study of Global Features for Time-Delayed Feedback Control 94

5.4.1 Analytical Bifurcation Analysis of One-Dimensional Maps 95

5.4.2 Dependence of Sub- and Supercritical Behavior on the Observable 98

5.4.3 Influence of the Coupling of the Control Force 99

5.5 Conclusion 101

Acknowledgments 102

Appendix A. Normal Form Reduction 103

Appendix B. Super- and Subcritical Hopf Bifurcation for Maps 106

References 106

6 Poincaré-Based Control of Delayed Measured Systems: Limitations and Improved Control 109
Jens Christian Claussen

6.1 Introduction 109

6.1.1 The Delay Problem–Time-Discrete Case 109

6.1.2 Experimental Setups with Delay 111

6.2 Ott-Grebogi-Yorke (OGY) Control 112

6.3 Limitations of Unmodified Control and Simple Improved Control Schemes 113

6.3.1 Limitations of Unmodified OGY Control in the Presence of Delay 113

6.3.2 Stability Diagrams Derived by the Jury Criterion 116

6.3.3 Stabilizing Unknown Fixed Points: Limitations of Unmodified Difference Control 116

6.3.4 Rhythmic Control Schemes: Rhythmic OGY Control 119

6.3.5 Rhythmic Difference Control 120

6.3.6 A Simple Memory Control Scheme: Using State Space Memory 122

6.4 Optimal Improved Control Schemes 123

6.4.1 Linear Predictive Logging Control (LPLC) 123

6.4.2 Nonlinear Predictive Logging Control 124

6.4.3 Stabilization of Unknown Fixed Points: Memory Difference Control (mdc) 125

6.5 Summary 126

References 127

7 Nonlinear and Adaptive Control of Chaos 129
Alexander Fradkov and Alexander Pogromsky

7.1 Introduction 129

7.2 Chaos and Control: Preliminaries 130

7.2.1 Definitions of Chaos 130

7.2.2 Models of Controlled Systems 131

7.2.3 Control Goals 132

7.3 Methods of Nonlinear Control 134

7.3.1 Gradient Method 135

7.3.2 Speed-Gradient Method 136

7.3.3 Feedback Linearization 141

7.3.4 Other Methods 142

7.3.5 Gradient Control of the Hénon System 144

7.3.6 Feedback Linearization Control of the Lorenz System 146

7.3.7 Speed-Gradient Stabilization of the Equilibrium Point for the Thermal Convection Loop Model 147

7.4 Adaptive Control 148

7.4.1 General Definitions 148

7.4.2 Adaptive Master-Slave Synchronization of Rössler Systems 149

7.5 Other Problems 154

7.6 Conclusions 155

Acknowledgment 155

References 156

Part II Controlling Space-time Chaos

8 Localized Control of Spatiotemporal Chaos 161
Roman O. Grigoriev and Andreas Handel

8.1 Introduction 161

8.1.1 Empirical Control 163

8.1.2 Model-Based Control 164

8.2 Symmetry and the Minimal Number of Sensors/Actuators 167

8.3 Nonnormality and Noise Amplification 170

8.4 Nonlinearity and the Critical Noise Level 175

8.5 Conclusions 177

References 177

9 Controlling Spatiotemporal Chaos: The Paradigm of the Complex Ginzburg-Landau Equation 181
Stefano Boccaletti and Jean Bragard

9.1 Introduction 181

9.2 The Complex Ginzburg-Landau Equation 183

9.2.1 Dynamics Characterization 185

9.3 Control of the CGLE 187

9.4 Conclusions and Perspectives 192

Acknowledgment 193

References 193

10 Multiple Delay Feedback Control 197
Alexander Ahlborn and Ulrich Parlitz

10.1 Introduction 197

10.2 Multiple Delay Feedback Control 198

10.2.1 Linear Stability Analysis 199

10.2.2 Example: Colpitts Oscillator 200

10.2.3 Comparison with High-Pass Filter and PD Controller 203

10.2.4 Transfer Function of MDFC 204

10.3 From Multiple Delay Feedback Control to Notch Filter Feedback 206

10.4 Controllability Criteria 208

10.4.1 Multiple Delay Feedback Control 209

10.4.2 Notch Filter Feedback and High-Pass Filter 210

10.5 Laser Stabilization Using MDFC and NFF 211

10.6 Controlling Spatiotemporal Chaos 213

10.6.1 The Ginzburg-Landau Equation 213

10.6.2 Controlling Traveling Plane Waves 214

10.6.3 Local Feedback Control 215

10.7 Conclusion 218

References 219

Part III Controlling Noisy Motion

11 Control of Noise-Induced Dynamics 223
Natalia B. Janson, Alexander G. Balanov, and Eckehard Schöll

11.1 Introduction 223

11.2 Noise-Induced Oscillations Below Andronov-Hopf Bifurcation and their Control 226

11.2.1 Weak Noise and Control: Correlation Function 228

11.2.2 Weak Noise and No Control: Correlation Time and Spectrum 229

11.2.3 Weak Noise and Control: Correlation Time 231

11.2.4 Weak Noise and Control: Spectrum 235

11.2.5 Any Noise and No Control: Correlation Time 236

11.2.6 Any Noise and Control: Correlation Time and Spectrum 238

11.2.7 So, What Can We Control? 240

11.3 Noise-Induced Oscillations in an Excitable System and their Control 241

11.3.1 Coherence Resonance in the Fitz Hugh-Nagumo System 243

11.3.2 Correlation Time and Spectrum when Feedback is Applied 244

11.3.3 Control of Synchronization in Coupled Fitz Hugh-Nagumo Systems 245

11.3.4 What can We Control in an Excitable System? 246

11.4 Delayed Feedback Control of Noise-Induced Pulses in a Model of an Excitable Medium 247

11.4.1 Model Description 247

11.4.2 Characteristics of Noise-Induced Patterns 249

11.4.3 Control of Noise-Induced Patterns 251

11.4.4 Mechanisms of Delayed Feedback Control of the Excitable Medium 253

11.4.5 What Can Be Controlled in an Excitable Medium? 254

11.5 Delayed Feedback Control of Noise-Induced Patterns in a Globally Coupled Reaction–Diffusion Model 255

11.5.1 Spatiotemporal Dynamics in the Uncontrolled Deterministic System 256

11.5.2 Noise-Induced Patterns in the Uncontrolled System 258

11.5.3 Time-Delayed Feedback Control of Noise-Induced Patterns 260

11.5.4 Linear Modes of the Inhomogeneous Fixed Point 264

11.5.5 Delay-Induced Oscillatory Patterns 268

11.5.6 What Can Be Controlled in a Globally Coupled Reaction–Diffusion System? 269

11.6 Summary and Conclusions 270

Acknowledgments 270

References 270

12 Controlling Coherence of Noisy and Chaotic Oscillators by Delayed Feedback 275
Denis Goldobin, Michael Rosenblum, and Arkady Pikovsky

12.1 Control of Coherence: Numerical Results 276

12.1.1 Noisy Oscillator 276

12.1.2 Chaotic Oscillator 277

12.1.3 Enhancing Phase Synchronization 279

12.2 Theory of Coherence Control 279

12.2.1 Basic Phase Model 279

12.2.2 Noise-Free Case 280

12.2.3 Gaussian Approximation 280

12.2.4 Self-Consistent Equation for Diffusion Constant 282

12.2.5 Comparison of Theory and Numerics 283

12.3 Control of Coherence by Multiple Delayed Feedback 283

12.4 Conclusion 288

References 289

13 Resonances Induced by the Delay Time in Nonlinear Autonomous Oscillators with Feedback 291
Cristina Masoller

Acknowledgment 298

References 299

Part IV Communicating with Chaos, Chaos Synchronization

14 Secure Communication with Chaos Synchronization 303
Wolfgang Kinzel and Ido Kanter

14.1 Introduction 303

14.2 Synchronization of Chaotic Systems 304

14.3 Coding and Decoding Secret Messages in Chaotic Signals 309

14.4 Analysis of the Exchanged Signal 311

14.5 Neural Cryptography 313

14.6 Public Key Exchange by Mutual Synchronization 315

14.7 Public Keys by Asymmetric Attractors 318

14.8 Mutual Chaos Pass Filter 319

14.9 Discussion 321

References 323

15 Noise Robust Chaotic Systems 325
Thomas L. Carroll

15.1 Introduction 325

15.2 Chaotic Synchronization 326

15.3 2-Frequency Self-Synchronizing Chaotic Systems 326

15.3.1 Simple Maps 326

15.4 2-Frequency Synchronization in Flows 329

15.4.1 2-Frequency Additive Rössler 329

15.4.2 Parameter Variation and Periodic Orbits 332

15.4.3 Unstable Periodic Orbits 333

15.4.4 Floquet Multipliers 334

15.4.5 Linewidths 335

15.5 Circuit Experiments 336

15.5.1 Noise Effects 338

15.6 Communication Simulations 338

15.7 Multiplicative Two-Frequency Rössler Circuit 341

15.8 Conclusions 346

References 346

16 Nonlinear Communication Strategies 349
Henry D.I. Abarbanel

16.1 Introduction 349

16.1.1 Secrecy, Encryption, and Security? 350

16.2 Synchronization 351

16.3 Communicating Using Chaotic Carriers 353

16.4 Two Examples from Optical Communication 355

16.4.1 Rare-Earth-Doped Fiber Amplifier Laser 355

16.4.2 Time Delay Optoelectronic Feedback Semiconductor Laser 357

16.5 Chaotic Pulse Position Communication 359

16.6 Why Use Chaotic Signals at All? 362

16.7 Undistorting the Nonlinear Effects of the Communication Channel 363

16.8 Conclusions 366

References 367

17 Synchronization and Message Transmission for Networked Chaotic Optical Communications 369
K. Alan Shore, Paul S. Spencer, and Ilestyn Pierce

17.1 Introduction 369

17.2 Synchronization and Message Transmission 370

17.3 Networked Chaotic Optical Communication 372

17.3.1 Chaos Multiplexing 373

17.3.2 Message Relay 373

17.3.3 Message Broadcasting 374

17.4 Summary 376

Acknowledgments 376

References 376

18 Feedback Control Principles for Phase Synchronization 379
Vladimir N. Belykh, Grigory V. Osipov, and Jürgen Kurths

18.1 Introduction 379

18.2 General Principles of Automatic Synchronization 381

18.3 Two Coupled Poincaré Systems 384

18.4 Coupled van der Pol and Rössler Oscillators 386

18.5 Two Coupled Rössler Oscillators 389

18.6 Coupled Rössler and Lorenz Oscillators 391

18.7 Principles of Automatic Synchronization in Networks of Coupled Oscillators 393

18.8 Synchronization of Locally Coupled Regular Oscillators 395

18.9 Synchronization of Locally Coupled Chaotic Oscillators 397

18.10 Synchronization of Globally Coupled Chaotic Oscillators 399

18.11 Conclusions 401

References 401

Part V Applications to Optics

19 Controlling Fast Chaos in Optoelectronic Delay Dynamical Systems 407
Lucas Illing, Daniel J. Gauthier, and Jonathan N. Blakely

19.1 Introduction 407

19.2 Control-Loop Latency: A Simple Example 408

19.3 Controlling Fast Systems 412

19.4 A Fast Optoelectronic Chaos Generator 415

19.5 Controlling the Fast Optoelectronic Device 419

19.6 Outlook 423

Acknowledgment 424

References 424

20 Control of Broad-Area Laser Dynamics with Delayed Optical Feedback 427
Nicoleta Gaciu, Edeltraud Gehrig, and Ortwin Hess

20.1 Introduction: Spatiotemporally Chaotic Semiconductor Lasers 427

20.2 Theory: Two-Level Maxwell-Bloch Equations 429

20.3 Dynamics of the Solitary Laser 432

20.4 Detection of Spatiotemporal Complexity 433

20.4.1 Reduction of the Number of Modes by Coherent Injection 433

20.4.2 Pulse-Induced Mode Synchronization 435

20.5 Self-Induced Stabilization and Control with Delayed Optical Feedback 438

20.5.1 Influence of Delayed Optical Feedback 439

20.5.2 Influence of the Delay Time 440

20.5.3 Spatially Structured Delayed Optical Feedback Control 444

20.5.4 Filtered Spatially Structured Delayed Optical Feedback 449

20.6 Conclusions 451

References 453

21 Noninvasive Control of Semiconductor Lasers by Delayed Optical Feedback 455
Hans-Jürgen Wünsche, Sylvia Schikora, and Fritz Henneberger

21.1 The Role of the Optical Phase 456

21.2 Generic Linear Model 459

21.3 Generalized Lang-Kobayashi Model 461

21.4 Experiment 462

21.4.1 The Integrated Tandem Laser 463

21.4.2 Design of the Control Cavity 464

21.4.3 Maintaining Resonance 465

21.4.4 Latency and Coupling Strength 465

21.4.5 Results of the Control Experiment 466

21.5 Numerical Simulation 468

21.5.1 Traveling-Wave Model 468

21.5.2 Noninvasive Control Beyond a Hopf Bifurcation 470

21.5.3 Control Dynamics 470

21.5.4 Variation of the Control Parameters 471

21.6 Conclusions 473

Acknowledgment 473

References 473

22 Chaos and Control in Semiconductor Lasers 475
Junji Ohtsubo

22.1 Introduction 475

22.2 Chaos in Semiconductor Lasers 476

22.2.1 Laser Chaos 476

22.2.2 Optical Feedback Effects in Semiconductor Lasers 478

22.2.3 Chaotic Effects in Newly Developed Semiconductor Lasers 480

22.3 Chaos Control in Semiconductor Lasers 485

22.4 Control in Newly Developed Semiconductor Lasers 494

22.5 Conclusions 497

References 498

23 From Pattern Control to Synchronization: Control Techniques in Nonlinear Optical Feedback Systems 501
Björn Gütlich and Cornelia Denz

23.1 Control Methods for Spatiotemporal Systems 502

23.2 Optical Single-Feedback Systems 503

23.2.1 A Simplified Single-Feedback Model System 504

23.2.2 The Photorefractive Single-Feedback System – Coherent Nonlinearity 506

23.2.3 Theoretical Description of the Photorefractive Single-Feedback System 508

23.2.4 Linear Stability Analysis 509

23.2.5 The LCLV Single-Feedback System – Incoherent Nonlinearity 510

23.2.6 Phase-Only Mode 511

23.2.7 Polarization Mode 513

23.2.8 Dissipative Solitons in the LCLV Feedback System 513

23.3 Spatial Fourier Control 514

23.3.1 Experimental Determination of Marginal Instability 516

23.3.2 Stabilization of Unstable Pattern 517

23.3.3 Direct Fourier Filtering 518

23.3.4 Positive Fourier Control 518

23.3.5 Noninvasive Fourier Control 519

23.4 Real-Space Control 520

23.4.1 Invasive Forcing 520

23.4.2 Positioning of Localized States 522

23.4.3 System Homogenization 522

23.4.4 Static Positioning 523

23.4.5 Addressing and Dynamic Positioning 523

23.5 Spatiotemporal Synchronization 524

23.5.1 Spatial Synchronization of Periodic Pattern 524

23.5.2 Unidirectional Synchronization of Two LCLV Systems 525

23.5.3 Synchronization of Spatiotemporal Complexity 526

23.6 Conclusions and Outlook 527

References 528

Part VI Applications to Electronic Systems

24 Delayed-Feedback Control of Chaotic Spatiotemporal Patterns in Semiconductor Nanostructures 533
Eckehard Schöll

24.1 Introduction 533

24.2 Control of Chaotic Domain and Front Patterns in Superlattices 536

24.3 Control of Chaotic Spatiotemporal Oscillations in Resonant Tunneling Diodes 544

24.4 Conclusions 553

Acknowledgments 554

References 554

25 Observing Global Properties of Time-Delayed Feedback Control in Electronic Circuits 559
Hartmut Benner, Chol-Ung Choe, Klaus Höhne, Clemens von Loewenich, Hiroyuki Shirahama, and Wolfram Just

25.1 Introduction 559

25.2 Discontinuous Transitions for Extended Time-Delayed Feedback Control 560

25.2.1 Theoretical Considerations 560

25.2.2 Experimental Setup 561

25.2.3 Observation of Bistability 562

25.2.4 Basin of Attraction 564

25.3 Controlling Torsion-Free Unstable Orbits 565

25.3.1 Applying the Concept of an Unstable Controller 567

25.3.2 Experimental Design of an Unstable van der Pol Oscillator 567

25.3.3 Control Coupling and Basin of Attraction 569

25.4 Conclusions 572

References 573

26 Application of a Black Box Strategy to Control Chaos 575
Achim Kittel and Martin Popp

26.1 Introduction 575

26.2 The Model Systems 575

26.2.1 Shinriki Oscillator 576

26.2.2 Mackey-Glass Type Oscillator 577

26.3 The Controller 580

26.4 Results of the Application of the Controller to the Shinriki Oscillator 582

26.4.1 Spectroscopy of Unstable Periodic Orbits 584

26.5 Results of the Application of the Controller to the Mackey-Glass Oscillator 585

26.5.1 Spectroscopy of Unstable Periodic Orbits 587

26.6 Further Improvements 589

26.7 Conclusions 589

Acknowledgment 590

References 590

Part VII Applications to Chemical Reaction Systems

27 Feedback-Mediated Control of Hypermeandering Spiral Waves 593
Jan Schlesner, Vladimir Zykov, and Harald Engel

27.1 Introduction 593

27.2 The Fitz Hugh-Nagumo Model 594

27.3 Stabilization of Rigidly Rotating Spirals in the Hypermeandering Regime 596

27.4 Control of Spiral Wave Location in the Hypermeandering Regime 599

27.5 Discussion 605

References 606

28 Control of Spatiotemporal Chaos in Surface Chemical Reactions 609
Carsten Beta and Alexander S. Mikhailov

28.1 Introduction 609

28.2 The Catalytic CO Oxidation on Pt(110) 610

28.2.1 Mechanism 610

28.2.2 Modeling 611

28.2.3 Experimental Setup 612

28.3 Spatiotemporal Chaos in Catalytic CO Oxidation on Pt(110) 613

28.4 Control of Spatiotemporal Chaos by Global Delayed Feedback 615

28.4.1 Control of Turbulence in Catalytic CO Oxidation – Experimental 616

28.4.1.1 Control of Turbulence 617

28.4.1.2 Spatiotemporal Pattern Formation 618

28.4.2 Control of Turbulence in Catalytic CO Oxidation – Numerical Simulations 619

28.4.3 Control of Turbulence in Oscillatory Media – Theory 621

28.4.4 Time-Delay Autosynchronization 625

28.5 Control of Spatiotemporal Chaos by Periodic Forcing 628

Acknowledgment 630

References 630

29 Forcing and Feedback Control of Arrays of Chaotic Electrochemical Oscillators 633
István Z. Kiss and John L. Hudson

29.1 Introduction 633

29.2 Control of Single Chaotic Oscillator 634

29.2.1 Experimental Setup 634

29.2.2 Chaotic Ni Dissolution: Low-Dimensional, Phase Coherent Attractor 635

29.2.2.1 Unforced Chaotic Oscillator 635

29.2.2.2 Phase of the Unforced System 636

29.2.3 Forcing: Phase Synchronization and Intermittency 637

29.2.3.1 Forcing with X=x 0 637

29.2.3.2 Forcing with X 6ˆ X 0 638

29.2.4 Delayed Feedback: Tracking 638

29.3 Control of Small Assemblies of Chaotic Oscillators 640

29.4 Control of Oscillator Populations 642

29.4.1 Global Coupling 642

29.4.2 Periodic Forcing of Arrays of Chaotic Oscillators 643

29.4.3 Feedback on Arrays of Chaotic Oscillators 644

29.4.4 Feedback, Forcing, and Global Coupling: Order Parameter 645

29.4.5 Control of Complexity of a Collective Signal 646

29.5 Concluding Remarks 647

Acknowledgment 648

References 649

Part VIII Applications to Biology

30 Control of Synchronization in Oscillatory Neural Networks 653
Peter A. Tass, Christian Hauptmann, and Oleksandr V. Popovych

30.1 Introduction 653

30.2 Multisite Coordinated Reset Stimulation 654

30.3 Linear Multisite Delayed Feedback 662

30.4 Nonlinear Delayed Feedback 666

30.5 Reshaping Neural Networks 674

30.6 Discussion 676

References 678

31 Control of Cardiac Electrical Nonlinear Dynamics 683
Trine Krogh-Madsen, Peter N. Jordan, and David J. Christini

31.1 Introduction 683

31.2 Cardiac Electrophysiology 684

31.2.1 Restitution and Alternans 685

31.3 Cardiac Arrhythmias 686

31.3.1 Reentry 687

31.3.2 Ventricular Tachyarrhythmias 688

31.3.3 Alternans as an Arrhythmia Trigger 688

31.4 Current Treatment of Arrhythmias 689

31.4.1 Pharmacological Treatment 689

31.4.2 Implantable Cardioverter Defibrillators 689

31.4.3 Ablation Therapy 690

31.5 Alternans Control 691

31.5.1 Controlling Cellular Alternans 691

31.5.2 Control of Alternans in Tissue 692

31.5.3 Limitations of the DFC Algorithm in Alternans Control 693

31.5.4 Adaptive DI Control 694

31.6 Control of Ventricular Tachyarrhythmias 695

31.6.1 Suppression of Spiral Waves 696

31.6.2 Antitachycardia Pacing 696

31.6.3 Unpinning Spiral Waves 698

31.7 Conclusions and Prospects 699

References 700

32 Controlling Spatiotemporal Chaos and Spiral Turbulence in Excitable Media 703
Sitabhra Sinha and S. Sridhar

32.1 Introduction 703

32.2 Models of Spatiotemporal Chaos in Excitable Media 706

32.3 Global Control 708

32.4 Nonglobal Spatially Extended Control 711

32.4.1 Applying Control Over a Mesh 711

32.4.2 Applying Control Over an Array of Points 713

32.5 Local Control of Spatiotemporal Chaos 714

32.6 Discussion 716

Acknowledgments 717

References 718

Part IX Applications to Engineering

33 Nonlinear Chaos Control and Synchronization 721
Henri J. C. Huijberts and Henk Nijmeijer

33.1 Introduction 721

33.2 Nonlinear Geometric Control 721

33.2.1 Some Differential Geometric Concepts 722

33.2.2 Nonlinear Controllability 723

33.2.3 Chaos Control Through Feedback Linearization 728

33.2.4 Chaos Control Through Input–Output Linearization 732

33.3 Lyapunov Design 737

33.3.1 Lyapunov Stability and Lyapunov’s First Method 737

33.3.2 Lyapunov’s Direct Method 739

33.3.3 La Salle’s Invariance Principle 741

33.3.4 Examples 742

References 749

34 Electronic Chaos Controllers – From Theory to Applications 751
Maciej Ogorzałek

34.1 Introduction 751

34.1.1 Chaos Control 752

34.1.2 Fundamental Properties of Chaotic Systems and Goals of the Control 753

34.2 Requirements for Electronic Implementation of Chaos Controllers 754

34.3 Short Description of the OGY Technique 755

34.4 Implementation Problems for the OGY Method 757

34.4.1 Effects of Calculation Precision 758

34.4.2 Approximate Procedures for Finding Periodic Orbits 759

34.4.3 Effects of Time Delays 759

34.5 Occasional Proportional Feedback (Hunt’s) Controller 761

34.5.1 Improved Chaos Controller for Autonomous Circuits 763

34.6 Experimental Chaos Control Systems 765

34.6.1 Control of a Magnetoelastic Ribbon 765

34.6.2 Control of a Chaotic Laser 766

34.6.3 Chaos-Based Arrhythmia Suppression and Defibrillation 767

34.7 Conclusions 768

References 769

35 Chaos in Pulse-Width Modulated Control Systems 771
Zhanybai T. Zhusubaliyev and Erik Mosekilde

35.1 Introduction 771

35.2 DC/DC Converter with Pulse-Width Modulated Control 774

35.3 Bifurcation Analysis for the DC/DC Converter with One-Level Control 778

35.4 DC/DC Converter with Two-Level Control 781

35.5 Bifurcation Analysis for the DC/DC Converter with Two-Level Control 783

35.6 Conclusions 784

Acknowledgments 788

References 788

36 Transient Dynamics of Duffing System Under Time-Delayed Feedback Control: Global Phase Structure and Application to Engineering 793
Takashi Hikihara and Kohei Yamasue

36.1 Introduction 793

36.2 Transient Dynamics of Transient Behavior 794

36.2.1 Magnetoelastic Beam and Experimental Setup 794

36.2.2 Transient Behavior 795

36.3 Initial Function and Domain of Attraction 797

36.4 Persistence of Chaos 800

36.5 Application of TDFC to Nanoengineering 803

36.5.1 Dynamic Force Microscopy and its Dynamics 803

36.5.2 Application of TDFC 805

36.5.3 Extension of Operating Range 806

36.6 Conclusions 808

References 808

Subject Index 811

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.