Dynamics of Nonlinear Time-Delay Systems (eBook)

Muthusamy Lakshmanan, Dharmapuri Vijayan Senthilkumar (Autoren)

eBook Download: PDF

2011 | 2011
XVII, 313 Seiten
Springer Berlin (Verlag)
978-3-642-14938-2 (ISBN)

Infos

Synchronization of chaotic systems, a patently nonlinear phenomenon, has emerged as a highly active interdisciplinary research topic at the interface of physics, biology, applied mathematics and engineering sciences. In this connection, time-delay systems described by delay differential equations have developed as particularly
suitable tools for modeling specific dynamical systems. Indeed, time-delay is ubiquitous in many physical systems, for example due to finite
switching speeds of amplifiers in electronic circuits, finite lengths of vehicles in traffic flows, finite signal propagation times in biological networks and circuits, and quite generally whenever memory effects are relevant.
This monograph presents the basics of chaotic time-delay systems and their synchronization with an emphasis on the effects of time-delay feedback which give rise to new collective dynamics.
Special attention is devoted to scalar chaotic/hyperchaotic time-delay
systems, and some higher order models, occurring in different branches of science and technology as well as to the synchronization of their coupled versions.

Last but not least, the presentation as a whole strives for a balance between the necessary mathematical description of the basics
and the detailed presentation of real-world applications.

Previous book by the same author. Nonlinear Dynamics - Integrability, Chaos and Patterns Series: Advanced Texts in Physics Lakshmanan, Muthusamy, Rajaseekar, Shanmuganathan 2003, XX, 619 p. 193 illus., Hardcover ISBN: 978-3-540-43908-0 Usually dispatched between 3 to 5 business days 159,95 € (original price unknown) Approx sales figures: 434 (ROW) + 200 (US).

Inhalt

Springer Complexity 1
Preface 6
Contents 9
Chapter 1 Delay Differential Equations 16
1.1 Introduction 16
1.1.1 DDE with Single Constant Delay 18
1.1.2 DDE with Discrete Delays 19
1.1.3 DDE with Distributed Delay 20
1.1.4 DDE with State-Dependent Delay 21
1.1.5 DDE with Time-Dependent Delay 21
1.2 Constructing the Solution for DDEs with Single Constant Delay 22
1.2.1 Linear Delay Differential Equation 23
1.2.2 Numerical Simulation of DDEs 25
1.2.3 Nonlinear Delay Differential Equations 26
1.3 Salient Features of Chaotic Time-Delay Systems 28
References 28
Chapter 2 Linear Stability and Bifurcation Analysis 31
2.1 Introduction 31
2.2 Linear Stability Analysis 31
2.2.1 Example: Linear Delay Differential Equation 33
2.3 A Geometric Approach to Study Stability 34
2.3.1 Example: Linear Delay Differential Equation 35
2.4 A General Approach to Determine Linear Stability of Equilibrium Points 36
2.4.1 Characteristic Equation 36
2.4.2 Stability Conditions 36
2.4.3 Stability Curves/Surfaces in the (,a,b) Parameter Space 37
2.4.4 Extension to Coupled DDEs/Complex Scalar DDEs 38
2.4.5 Bifurcation Analysis 39
2.4.6 Results of Stability Analysis 39
2.4.7 A Theorem on the Stability of Equilibrium Points 40
2.4.8 Example: Linear Delay Differential Equation 40
References 43
Chapter 3 Bifurcation and Chaos in Time-Delayed Piecewise Linear Dynamical System 44
3.1 Introduction 44
3.2 Simple Scalar First Order Piecewise Linear DDE 45
3.2.1 Fixed Points and Linear Stability 46
3.3 Numerical Study of the Single Scalar Piecewise Linear Time-Delay System 49
3.3.1 Dynamics in the Pseudospace 49
3.3.2 Transients 50
3.3.3 One and Two Parameter Bifurcation Diagrams 54
3.3.4 Lyapunov Exponents and Hyperchaotic Regimes 56
3.4 Experimental Realization using PSPICE Simulation 57
3.5 Stability Analysis and Chaotic Dynamics of Coupled DDEs 59
3.5.1 Fixed Points and Linear Stability 59
3.6 Numerical Analysis of the Coupled DDE 62
3.6.1 Transients 63
3.6.2 One and Two Parameter Bifurcation Diagrams 64
References 66
Chapter 4 A Few Other Interesting Chaotic Delay Differential Equations 68
4.1 Introduction 68
4.2 The Mackey-Glass System: A Typical Nonlinear DDE 68
4.2.1 Mackey-Glass Time-Delay System 68
4.2.2 Fixed Points and Linear Stability Analysis 69
4.2.3 Time-Delay =0 70
4.2.4 Time-Delay > 0
4.2.5 Numerical Simulation: Bifurcations and Chaos 75
4.2.6 Experimental Realization Using Electronic Circuit 77
4.3 Other Interesting Scalar Chaotic Time-Delay Systems 80
4.3.1 A Simple Chaotic Delay Differential Equation 80
4.3.2 Ikeda Time-Delay System 80
4.3.3 Scalar Time-Delay System with Polynomial Nonlinearity 82
4.3.4 Scalar Time-Delay System with Other Piecewise Linear Nonlinearities 83
4.3.5 Another Form of Scalar Time-Delay System 86
4.3.6 El Niño and the Delayed Action Oscillator 89
4.4 Coupled Chaotic Time-Delay Systems 91
4.4.1 Time-Delayed Chua's Circuit 91
4.4.2 Semiconductor Lasers 92
4.4.3 Neural Networks 94
References 95
Chapter 5 Implications of Delay Feedback: Amplitude Death and Other Effects 98
5.1 Introduction 98
5.2 Time-Delay Induced Amplitude Death 98
5.2.1 Theoretical Study: Single Oscillator 99
5.2.2 Experimental Study 102
5.3 Amplitude Death with Distributed Delay in Coupled Limit Cycle Oscillators 104
5.4 Amplitude Death in Coupled Chaotic Oscillators 106
5.5 Amplitude Death with Conjugate (Dissimilar) Coupling 109
5.6 Amplitude Death with Dynamic Coupling 111
5.7 Time-Delay Induced Bifurcations 114
5.8 Some Other Effects of Delay Feedback 115
References 116
Chapter 6 Recent Developments on Delay Feedback/Coupling: Complex Networks, Chimeras, Globally Clustered Chimeras and Synchronization 117
6.1 Introduction 117
6.2 Complex Networks 117
6.3 Chimera States in Delay Coupled Identical Oscillators 120
6.3.1 Discovery of Chimera States 120
6.3.2 Chimera States in Delay Coupled Systems 123
6.4 Chimera States in Delay Coupled Subpopulations: Globally Clustered States 125
6.5 Synchronization in Complex Networks with Delay 129
6.6 Controlling Using Time-Delay Feedback 130
6.6.1 Pyragas Time-Delay Feedback Control 131
6.6.2 Transient Behavior with Time-Delay Feedback 134
6.7 Further Developments 136
References 137
Chapter 7 Complete Synchronization of Chaotic Oscillations in Coupled Time-Delay Systems 139
7.1 Introduction 139
7.2 Complete Synchronization in Coupled Time-Delay Systems 141
7.3 Stability Using Krasovskii-Lyapunov Theory 142
7.4 Numerical Confirmation 145
7.4.1 Case 1 146
7.4.2 Case 2 146
7.4.3 Case 3 147
7.4.4 Case 4 147
7.5 Conclusion 147
References 148
Chapter 8 Transition from Anticipatory to Lag Synchronization via Complete Synchronization 151
8.1 Introduction 151
8.2 Coupled System and the General Stability Condition 151
8.3 Coupled Piecewise Linear Time-Delay System and Stability Condition: Transition from Anticipatory to Lag Synchronization 153
8.3.1 Anticipatory Synchronization for 2 < 1
8.3.2 Complete Synchronization for 2 = 1 158
8.3.3 Lag Synchronization for 2 > 1
8.3.4 Inverse Synchronizations 161
8.4 Transition from Anticipatory to Lag via Complete Synchronization: Mackey-Glass and Ikeda Systems 165
8.4.1 Anticipatory Synchronization for 2 < 1
8.4.2 Complete Synchronization for 2 = 1 170
8.4.3 Lag Synchronization for 2 > 1
8.5 Inverse Synchronizations: Mackey-Glass and Ikeda Systems 173
References 175
Chapter 9 Intermittency Transition to Generalized Synchronization 177
9.1 Introduction 177
9.2 Broad Range (Slow/Delayed) Intermittency Transition to GS for Linear Error Feedback Coupling of the Form (x1(t)-x2(t)) 178
9.3 Stability Condition 179
9.4 Approximate (Intermittent) Generalized Synchronization 180
9.5 Characterization of IGS 183
9.6 Narrow Range (Immediate) Intermittency Transition to GS for Linear Direct Feedback Coupling of the Form x1(t) 185
9.7 Broad Range Intermittency Transition to GS for Nonlinear Error Feedback Coupling of the Form (f(x1(t-2))-f(x2(t-2))) 190
9.8 Narrow Range Intermittency Transition to GS for Nonlinear Direct Feedback Coupling of the Form f(x1(t-2)) 193
9.9 Intermittency Transition to Generalized Synchronization: Mackey-Glass & Ikeda Systems
9.9.1 Broad Range Intermittency Transition to GS 198
9.9.2 Narrow Range Intermittency Transition to GS 202
References 211
Chapter 10 Transition from Phase to Generalized Synchronization 212
10.1 Introduction 212
10.2 Phase-Coherent and Non-phase-coherent Attractors 213
10.3 CPS in Chaotic Systems 214
10.4 CPS and Time-Delay Systems 216
10.5 CPS from Poincaré Surface of Section of the Transformed Attractor 218
10.6 CPS from Recurrence Quantification Analysis 221
10.7 CPS from the Lyapunov Exponents 224
10.8 Concept of Localized Sets 225
10.9 Transition from Phase to Generalized Synchronization: Mackey-Glass & Ikeda Systems
10.9.1 CPS from Poincaré Section of the Transformed Attractor 228
10.9.2 CPS from Recurrence Quantification Analysis 229
10.9.3 CPS from the Lyapunov Exponents 230
10.9.4 CPS in Coupled Ikeda Systems 232
10.10 Summary 236
References 236
Chapter 11 DTM Induced Oscillating Synchronization 238
11.1 Introduction 238
11.2 Estimation of the Effect of Delay Time Modulation 239
11.2.1 Filling Factor 239
11.2.2 Length of Polygon Line 241
11.2.3 Average Mutual Information 241
11.3 Coupled System and Stability Condition in the Presence of Delay Time Modulation 244
11.4 Oscillating Synchronization 246
11.5 Intermittent Anticipatory Synchronization 249
11.6 Complete Synchronization 252
11.7 Intermittent Lag Synchronization 253
11.8 Complex Oscillating Synchronization 256
11.9 DTM Induced Oscillating Synchronization: Mackey-Glass & Ikeda Systems
11.9.1 Coupled Mackey-Glass Systems 256
11.9.2 Coupled Ikeda Systems 257
11.10 Summary 259
References 260
Chapter 12 Exact Solutions of Certain Time Delay Systems: The Car-Following Models 262
12.1 Introduction 262
12.2 The Car-Following Models 262
12.3 The Newell Model 263
12.4 The tanh Car-Following Model 266
12.5 Other Developments 268
References 269
Appendix A Computing Lyapunov Exponents for Time-DelaySystems 270
Appendix B A Brief Introduction to Synchronizationin Chaotic Dynamical Systems 274
Appendix C Recurrence Analysis 289
Appendix D Some More Examples of DDEs 302
Glossary 309
Index 317

Produkt-Details

• Erscheint lt. Verlag: 4.1.2011
• Reihe/Serie: Springer Series in Synergetics
• Reihe/Serie: Springer Series in Synergetics
• Zusatzinfo: XVII, 313 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Literatur
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Astronomie / Astrophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Schlagworte: chaotic dynamical systems • Delay differential equations • delay feedback • Electronic circuits • in complex networks • recurrence plots and analysis • Synchronization • time delay systesm
• ISBN-10: 3-642-14938-3 / 3642149383
• ISBN-13: 978-3-642-14938-2 / 9783642149382