Handbook of Chaos Control

Handbook of Chaos Control......Page 4
Contents......Page 8
Preface......Page 24
List of Contributors......Page 26
Part I Basic Aspects and Extension of Methods......Page 34
1.1 Introduction......Page 36
1.2 The OGY Chaos Control......Page 39
1.3 Targeting–Steering Chaotic Trajectories......Page 41
1.3.1 Part I: Finding a Proper Trajectory......Page 42
1.3.2 Part II: Finding a Pseudo-Orbit Trajectory......Page 43
1.3.3 The Targeting Algorithm......Page 45
1.4.1 Controlling an Electronic Circuit......Page 46
1.4.2 Controlling a Complex System......Page 52
References......Page 59
2.1 Overview: Why Study Discrete Maps?......Page 62
2.2 Theme and Variations......Page 64
2.2.1 Rudimentary Time-Delay Feedback......Page 65
2.2.2 Extending the Domain of Control......Page 67
2.2.3 High-Dimensional Systems......Page 70
2.3 Robustness of Time-Delay Stabilization......Page 74
References......Page 77
3.1 Introduction......Page 80
3.2 Proportional Versus Delayed Feedback......Page 83
3.3 Controlling Periodic Orbits Arising from a Period Doubling Bifurcation......Page 86
3.3.1 Example: Controlling the Rössler System......Page 87
3.4.1 Problem Formulation and Averaged Equation......Page 90
3.4.2 Periodic Orbits of the Free System......Page 91
3.4.3 Linear Stability of the System Controlled by Delayed Feedback......Page 93
3.5 Controlling Torsion-Free Periodic Orbits......Page 96
3.5.1 Example: Controlling the Lorenz System at a Subcritical Hopf Bifurcation......Page 98
3.6 Conclusions......Page 101
References......Page 103
4.1 Introduction......Page 106
4.2 Mechanism of Stabilization......Page 107
4.3 Conditions on the Feedback Gain......Page 111
Appendix: Calculation of Floquet Exponents......Page 115
References......Page 116
5.1 Introduction......Page 118
5.2 A Comment on Control and Root Finding Algorithms......Page 121
5.3.1 The Transition from Super- to Subcritical Behavior......Page 124
5.3.2 Probing Basins of Attraction in Experiments......Page 126
5.4 A Case Study of Global Features for Time-Delayed Feedback Control......Page 127
5.4.1 Analytical Bifurcation Analysis of One-Dimensional Maps......Page 128
5.4.2 Dependence of Sub- and Supercritical Behavior on the Observable......Page 131
5.4.3 Influence of the Coupling of the Control Force......Page 132
5.5 Conclusion......Page 134
Acknowledgments......Page 135
Appendix A Normal Form Reduction......Page 136
References......Page 139
6.1.1 The Delay Problem–Time-Discrete Case......Page 142
6.1.2 Experimental Setups with Delay......Page 144
6.2 Ott-Grebogi-Yorke (OGY) Control......Page 145
6.3.1 Limitations of Unmodified OGY Control in the Presence of Delay......Page 146
6.3.3 Stabilizing Unknown Fixed Points: Limitations of Unmodified Difference Control......Page 149
6.3.4 Rhythmic Control Schemes: Rhythmic OGY Control......Page 152
6.3.5 Rhythmic Difference Control......Page 153
6.3.6 A Simple Memory Control Scheme: Using State Space Memory......Page 155
6.4.1 Linear Predictive Logging Control (LPLC)......Page 156
6.4.2 Nonlinear Predictive Logging Control......Page 157
6.4.3 Stabilization of Unknown Fixed Points: Memory Difference Control (MDC)......Page 158
6.5 Summary......Page 159
References......Page 160
7.1 Introduction......Page 162
7.2.1 Definitions of Chaos......Page 163
7.2.2 Models of Controlled Systems......Page 164
7.2.3 Control Goals......Page 165
7.3 Methods of Nonlinear Control......Page 167
7.3.1 Gradient Method......Page 168
7.3.2 Speed-Gradient Method......Page 169
7.3.3 Feedback Linearization......Page 174
7.3.4 Other Methods......Page 175
7.3.5 Gradient Control of the Hénon System......Page 177
7.3.6 Feedback Linearization Control of the Lorenz System......Page 179
7.3.7 Speed-Gradient Stabilization of the Equilibrium Point for the Thermal Convection Loop Model......Page 180
7.4.1 General Definitions......Page 181
7.4.2 Adaptive Master-Slave Synchronization of Rössler Systems......Page 182
7.5 Other Problems......Page 187
Acknowledgment......Page 188
References......Page 189
Part II Controlling Space-Time Chaos......Page 192
8.1 Introduction......Page 194
8.1.1 Empirical Control......Page 196
8.1.2 Model-Based Control......Page 197
8.2 Symmetry and the Minimal Number of Sensors/Actuators......Page 200
8.3 Nonnormality and Noise Amplification......Page 203
8.4 Nonlinearity and the Critical Noise Level......Page 208
References......Page 210
9.1 Introduction......Page 214
9.2 The Complex Ginzburg-Landau Equation......Page 216
9.2.1 Dynamics Characterization......Page 218
9.3 Control of the CGLE......Page 220
9.4 Conclusions and Perspectives......Page 225
References......Page 226
10.1 Introduction......Page 230
10.2 Multiple Delay Feedback Control......Page 231
10.2.1 Linear Stability Analysis......Page 232
10.2.2 Example: Colpitts Oscillator......Page 233
10.2.3 Comparison with High-Pass Filter and PD Controller......Page 236
10.2.4 Transfer Function of MDFC......Page 237
10.3 From Multiple Delay Feedback Control to Notch Filter Feedback......Page 239
10.4 Controllability Criteria......Page 241
10.4.1 Multiple Delay Feedback Control......Page 242
10.4.2 Notch Filter Feedback and High-Pass Filter......Page 243
10.5 Laser Stabilization Using MDFC and NFF......Page 244
10.6.1 The Ginzburg-Landau Equation......Page 246
10.6.2 Controlling Traveling Plane Waves......Page 247
10.6.3 Local Feedback Control......Page 248
10.7 Conclusion......Page 251
References......Page 252
Part III Controlling Noisy Motion......Page 254
11.1 Introduction......Page 256
11.2 Noise-Induced Oscillations Below Andronov-Hopf Bifurcation and their Control......Page 259
11.2.1 Weak Noise and Control: Correlation Function......Page 261
11.2.2 Weak Noise and No Control: Correlation Time and Spectrum......Page 262
11.2.3 Weak Noise and Control: Correlation Time......Page 264
11.2.4 Weak Noise and Control: Spectrum......Page 268
11.2.5 Any Noise and No Control: Correlation Time......Page 269
11.2.6 Any Noise and Control: Correlation Time and Spectrum......Page 271
11.2.7 So, What Can We Control?......Page 273
11.3 Noise-Induced Oscillations in an Excitable System and their Control......Page 274
11.3.1 Coherence Resonance in the FitzHugh-Nagumo System......Page 276
11.3.2 Correlation Time and Spectrum when Feedback is Applied......Page 277
11.3.3 Control of Synchronization in Coupled FitzHugh-Nagumo Systems......Page 278
11.3.4 What can We Control in an Excitable System?......Page 279
11.4.1 Model Description......Page 280
11.4.2 Characteristics of Noise-Induced Patterns......Page 282
11.4.3 Control of Noise-Induced Patterns......Page 284
11.4.4 Mechanisms of Delayed Feedback Control of the Excitable Medium......Page 286
11.4.5 What Can Be Controlled in an Excitable Medium?......Page 287
11.5 Delayed Feedback Control of Noise-Induced Patterns in a Globally Coupled Reaction–Diffusion Model......Page 288
11.5.1 Spatiotemporal Dynamics in the Uncontrolled Deterministic System......Page 289
11.5.2 Noise-Induced Patterns in the Uncontrolled System......Page 291
11.5.3 Time-Delayed Feedback Control of Noise-Induced Patterns......Page 293
11.5.4 Linear Modes of the Inhomogeneous Fixed Point......Page 297
11.5.5 Delay-Induced Oscillatory Patterns......Page 301
11.5.6 What Can Be Controlled in a Globally Coupled Reaction–Diffusion System?......Page 302
References......Page 303
12 Controlling Coherence of Noisy and Chaotic Oscillators by Delayed Feedback......Page 308
12.1.1 Noisy Oscillator......Page 309
12.1.2 Chaotic Oscillator......Page 310
12.2.1 Basic Phase Model......Page 312
12.2.3 Gaussian Approximation......Page 313
12.2.4 Self-Consistent Equation for Diffusion Constant......Page 315
12.3 Control of Coherence by Multiple Delayed Feedback......Page 316
12.4 Conclusion......Page 321
References......Page 322
13 Resonances Induced by the Delay Time in Nonlinear Autonomous Oscillators with Feedback......Page 324
Acknowledgment......Page 331
References......Page 332
Part IV Communicating with Chaos, Chaos Synchronization......Page 334
14.1 Introduction......Page 336
14.2 Synchronization of Chaotic Systems......Page 337
14.3 Coding and Decoding Secret Messages in Chaotic Signals......Page 342
14.4 Analysis of the Exchanged Signal......Page 344
14.5 Neural Cryptography......Page 346
14.6 Public Key Exchange by Mutual Synchronization......Page 348
14.7 Public Keys by Asymmetric Attractors......Page 351
14.8 Mutual Chaos Pass Filter......Page 352
14.9 Discussion......Page 354
References......Page 356
15.1 Introduction......Page 358
15.3.1 Simple Maps......Page 359
15.4.1 2-Frequency Additive Rössler......Page 362
15.4.2 Parameter Variation and Periodic Orbits......Page 365
15.4.3 Unstable Periodic Orbits......Page 366
15.4.4 Floquet Multipliers......Page 367
15.4.5 Linewidths......Page 368
15.5 Circuit Experiments......Page 369
15.6 Communication Simulations......Page 371
15.7 Multiplicative Two-Frequency Rössler Circuit......Page 374
References......Page 379
16.1 Introduction......Page 382
16.1.1 Secrecy, Encryption, and Security?......Page 383
16.2 Synchronization......Page 384
16.3 Communicating Using Chaotic Carriers......Page 386
16.4.1 Rare-Earth-Doped Fiber Amplifier Laser......Page 388
16.4.2 Time Delay Optoelectronic Feedback Semiconductor Laser......Page 390
16.5 Chaotic Pulse Position Communication......Page 392
16.6 Why Use Chaotic Signals at All?......Page 395
16.7 Undistorting the Nonlinear Effects of the Communication Channel......Page 396
16.8 Conclusions......Page 399
References......Page 400
17.1 Introduction......Page 402
17.2 Synchronization and Message Transmission......Page 403
17.3 Networked Chaotic Optical Communication......Page 405
17.3.2 Message Relay......Page 406
17.3.3 Message Broadcasting......Page 407
References......Page 409
18.1 Introduction......Page 412
18.2 General Principles of Automatic Synchronization......Page 414
18.3 Two Coupled Poincaré Systems......Page 417
18.4 Coupled van der Pol and Rössler Oscillators......Page 419
18.5 Two Coupled Rössler Oscillators......Page 422
18.6 Coupled Rössler and Lorenz Oscillators......Page 424
18.7 Principles of Automatic Synchronization in Networks of Coupled Oscillators......Page 426
18.8 Synchronization of Locally Coupled Regular Oscillators......Page 428
18.9 Synchronization of Locally Coupled Chaotic Oscillators......Page 430
18.10 Synchronization of Globally Coupled Chaotic Oscillators......Page 432
References......Page 434
Part V Applications to Optics......Page 438
19.1 Introduction......Page 440
19.2 Control-Loop Latency: A Simple Example......Page 441
19.3 Controlling Fast Systems......Page 445
19.4 A Fast Optoelectronic Chaos Generator......Page 448
19.5 Controlling the Fast Optoelectronic Device......Page 452
19.6 Outlook......Page 456
References......Page 457
20.1 Introduction: Spatiotemporally Chaotic Semiconductor Lasers......Page 460
20.2 Theory: Two-Level Maxwell-Bloch Equations......Page 462
20.3 Dynamics of the Solitary Laser......Page 465
20.4.1 Reduction of the Number of Modes by Coherent Injection......Page 466
20.4.2 Pulse-Induced Mode Synchronization......Page 468
20.5 Self-Induced Stabilization and Control with Delayed Optical Feedback......Page 471
20.5.1 Influence of Delayed Optical Feedback......Page 472
20.5.2 Influence of the Delay Time......Page 473
20.5.3 Spatially Structured Delayed Optical Feedback Control......Page 477
20.5.4 Filtered Spatially Structured Delayed Optical Feedback......Page 482
20.6 Conclusions......Page 484
References......Page 486
21 Noninvasive Control of Semiconductor Lasers by Delayed Optical Feedback......Page 488
21.1 The Role of the Optical Phase......Page 489
21.2 Generic Linear Model......Page 492
21.3 Generalized Lang-Kobayashi Model......Page 494
21.4 Experiment......Page 495
21.4.1 The Integrated Tandem Laser......Page 496
21.4.2 Design of the Control Cavity......Page 497
21.4.4 Latency and Coupling Strength......Page 498
21.4.5 Results of the Control Experiment......Page 499
21.5.1 Traveling-Wave Model......Page 501
21.5.3 Control Dynamics......Page 503
21.5.4 Variation of the Control Parameters......Page 504
References......Page 506
22.1 Introduction......Page 508
22.2.1 Laser Chaos......Page 509
22.2.2 Optical Feedback Effects in Semiconductor Lasers......Page 511
22.2.3 Chaotic Effects in Newly Developed Semiconductor Lasers......Page 513
22.3 Chaos Control in Semiconductor Lasers......Page 518
22.4 Control in Newly Developed Semiconductor Lasers......Page 527
22.5 Conclusions......Page 530
References......Page 531
23 From Pattern Control to Synchronization: Control Techniques in Nonlinear Optical Feedback Systems......Page 534
23.1 Control Methods for Spatiotemporal Systems......Page 535
23.2 Optical Single-Feedback Systems......Page 536
23.2.1 A Simplified Single-Feedback Model System......Page 537
23.2.2 The Photorefractive Single-Feedback System – Coherent Nonlinearity......Page 539
23.2.3 Theoretical Description of the Photorefractive Single-Feedback System......Page 541
23.2.4 Linear Stability Analysis......Page 542
23.2.5 The LCLV Single-Feedback System – Incoherent Nonlinearity......Page 543
23.2.6 Phase-Only Mode......Page 544
23.2.8 Dissipative Solitons in the LCLV Feedback System......Page 546
23.3 Spatial Fourier Control......Page 547
23.3.1 Experimental Determination of Marginal Instability......Page 549
23.3.2 Stabilization of Unstable Pattern......Page 550
23.3.4 Positive Fourier Control......Page 551
23.3.5 Noninvasive Fourier Control......Page 552
23.4.1 Invasive Forcing......Page 553
23.4.3 System Homogenization......Page 555
23.4.5 Addressing and Dynamic Positioning......Page 556
23.5.1 Spatial Synchronization of Periodic Pattern......Page 557
23.5.2 Unidirectional Synchronization of Two LCLV Systems......Page 558
23.5.3 Synchronization of Spatiotemporal Complexity......Page 559
23.6 Conclusions and Outlook......Page 560
References......Page 561
Part VI Applications to Electronic Systems......Page 564
24.1 Introduction......Page 566
24.2 Control of Chaotic Domain and Front Patterns in Superlattices......Page 569
24.3 Control of Chaotic Spatiotemporal Oscillations in Resonant Tunneling Diodes......Page 577
24.4 Conclusions......Page 586
References......Page 587
25.1 Introduction......Page 592
25.2.1 Theoretical Considerations......Page 593
25.2.2 Experimental Setup......Page 594
25.2.3 Observation of Bistability......Page 595
25.2.4 Basin of Attraction......Page 597
25.3 Controlling Torsion-Free Unstable Orbits......Page 598
25.3.2 Experimental Design of an Unstable van der Pol Oscillator......Page 600
25.3.3 Control Coupling and Basin of Attraction......Page 602
25.4 Conclusions......Page 605
References......Page 606
26.2 The Model Systems......Page 608
26.2.1 Shinriki Oscillator......Page 609
26.2.2 Mackey-Glass Type Oscillator......Page 610
26.3 The Controller......Page 613
26.4 Results of the Application of the Controller to the Shinriki Oscillator......Page 615
26.4.1 Spectroscopy of Unstable Periodic Orbits......Page 617
26.5 Results of the Application of the Controller to the Mackey-Glass Oscillator......Page 618
26.5.1 Spectroscopy of Unstable Periodic Orbits......Page 620
26.7 Conclusions......Page 622
References......Page 623
Part VII Applications to Chemical Reaction Systems......Page 624
27.1 Introduction......Page 626
27.2 The FitzHugh-Nagumo Model......Page 627
27.3 Stabilization of Rigidly Rotating Spirals in the Hypermeandering Regime......Page 629
27.4 Control of Spiral Wave Location in the Hypermeandering Regime......Page 632
27.5 Discussion......Page 638
References......Page 639
28.1 Introduction......Page 642
28.2.1 Mechanism......Page 643
28.2.2 Modeling......Page 644
28.2.3 Experimental Setup......Page 645
28.3 Spatiotemporal Chaos in Catalytic CO Oxidation on Pt(110)......Page 646
28.4 Control of Spatiotemporal Chaos by Global Delayed Feedback......Page 648
28.4.1 Control of Turbulence in Catalytic CO Oxidation – Experimental......Page 649
28.4.1.1 Control of Turbulence......Page 650
28.4.1.2 Spatiotemporal Pattern Formation......Page 651
28.4.2 Control of Turbulence in Catalytic CO Oxidation – Numerical Simulations......Page 652
28.4.3 Control of Turbulence in Oscillatory Media – Theory......Page 654
28.4.4 Time Delay Autosynchronization......Page 658
28.5 Control of Spatiotemporal Chaos by Periodic Forcing......Page 661
References......Page 663
29.1 Introduction......Page 666
29.2.1 Experimental Setup......Page 667
29.2.2.1 Unforced Chaotic Oscillator......Page 668
29.2.2.2 Phase of the Unforced System......Page 669
29.2.3.1 Forcing with Ω = ω(0)......Page 670
29.2.4 Delayed Feedback: Tracking......Page 671
29.3 Control of Small Assemblies of Chaotic Oscillators......Page 673
29.4.1 Global Coupling......Page 675
29.4.2 Periodic Forcing of Arrays of Chaotic Oscillators......Page 676
29.4.3 Feedback on Arrays of Chaotic Oscillators......Page 677
29.4.4 Feedback, Forcing, and Global Coupling: Order Parameter......Page 678
29.4.5 Control of Complexity of a Collective Signal......Page 679
29.5 Concluding Remarks......Page 680
Acknowledgment......Page 681
References......Page 682
Part VIII Applications to Biology......Page 684
30.1 Introduction......Page 686
30.2 Multisite Coordinated Reset Stimulation......Page 687
30.3 Linear Multisite Delayed Feedback......Page 695
30.4 Nonlinear Delayed Feedback......Page 699
30.5 Reshaping Neural Networks......Page 707
30.6 Discussion......Page 709
References......Page 711
31.1 Introduction......Page 716
31.2 Cardiac Electrophysiology......Page 717
31.2.1 Restitution and Alternans......Page 718
31.3 Cardiac Arrhythmias......Page 719
31.3.1 Reentry......Page 720
31.3.3 Alternans as an Arrhythmia Trigger......Page 721
31.4.2 Implantable Cardioverter Defibrillators......Page 722
31.4.3 Ablation Therapy......Page 723
31.5.1 Controlling Cellular Alternans......Page 724
31.5.2 Control of Alternans in Tissue......Page 725
31.5.3 Limitations of the DFC Algorithm in Alternans Control......Page 726
31.5.4 Adaptive DI Control......Page 727
31.6 Control of Ventricular Tachyarrhythmias......Page 728
31.6.2 Antitachycardia Pacing......Page 729
31.6.3 Unpinning Spiral Waves......Page 731
31.7 Conclusions and Prospects......Page 732
References......Page 733
32.1 Introduction......Page 736
32.2 Models of Spatiotemporal Chaos in Excitable Media......Page 739
32.3 Global Control......Page 741
32.4.1 Applying Control Over a Mesh......Page 744
32.4.2 Applying Control Over an Array of Points......Page 746
32.5 Local Control of Spatiotemporal Chaos......Page 747
32.6 Discussion......Page 749
Acknowledgments......Page 750
References......Page 751
Part IX Applications to Engineering......Page 752
33.2 Nonlinear Geometric Control......Page 754
33.2.1 Some Differential Geometric Concepts......Page 755
33.2.2 Nonlinear Controllability......Page 756
33.2.3 Chaos Control Through Feedback Linearization......Page 761
33.2.4 Chaos Control Through Input–Output Linearization......Page 765
33.3.1 Lyapunov Stability and Lyapunov’s First Method......Page 770
33.3.2 Lyapunov’s Direct Method......Page 772
33.3.3 LaSalle’s Invariance Principle......Page 774
33.3.4 Examples......Page 775
References......Page 782
34.1 Introduction......Page 784
34.1.1 Chaos Control......Page 785
34.1.2 Fundamental Properties of Chaotic Systems and Goals of the Control......Page 786
34.2 Requirements for Electronic Implementation of Chaos Controllers......Page 787
34.3 Short Description of the OGY Technique......Page 788
34.4 Implementation Problems for the OGY Method......Page 790
34.4.1 Effects of Calculation Precision......Page 791
34.4.3 Effects of Time Delays......Page 792
34.5 Occasional Proportional Feedback (Hunt's) Controller......Page 794
34.5.1 Improved Chaos Controller for Autonomous Circuits......Page 796
34.6.1 Control of a Magnetoelastic Ribbon......Page 798
34.6.2 Control of a Chaotic Laser......Page 799
34.6.3 Chaos-Based Arrhythmia Suppression and Defibrillation......Page 800
34.7 Conclusions......Page 801
References......Page 802
35.1 Introduction......Page 804
35.2 DC/DC Converter with Pulse-Width Modulated Control......Page 807
35.3 Bifurcation Analysis for the DC/DC Converter with One-Level Control......Page 811
35.4 DC/DC Converter with Two-Level Control......Page 814
35.5 Bifurcation Analysis for the DC/DC Converter with Two-Level Control......Page 816
35.6 Conclusions......Page 821
References......Page 823
36.1 Introduction......Page 826
36.2.1 Magnetoelastic Beam and Experimental Setup......Page 827
36.2.2 Transient Behavior......Page 828
36.3 Initial Function and Domain of Attraction......Page 830
36.4 Persistence of Chaos......Page 833
36.5.1 Dynamic Force Microscopy and its Dynamics......Page 836
36.5.2 Application of TDFC......Page 838
36.5.3 Extension of Operating Range......Page 839
References......Page 841
Subject Index......Page 844