Lakshmanan / Senthilkumar Dynamics of Nonlinear Time-Delay Systems

1. Auflage 2011
ISBN: 978-3-642-14938-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 313 Seiten

Reihe: Springer Series in Synergetics

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).

Autoren/Hrsg.

Lakshmanan, Muthusamy

Senthilkumar, Dharmapuri Vijayan

Inhaltsverzeichnis

1;Springer Complexity;1
2;Preface;6
3;Contents;9
4;Chapter 1 Delay Differential Equations;16
4.1;1.1 Introduction;16
4.1.1;1.1.1 DDE with Single Constant Delay;18
4.1.2;1.1.2 DDE with Discrete Delays;19
4.1.3;1.1.3 DDE with Distributed Delay;20
4.1.4;1.1.4 DDE with State-Dependent Delay;21
4.1.5;1.1.5 DDE with Time-Dependent Delay;21
4.2;1.2 Constructing the Solution for DDEs with Single Constant Delay;22
4.2.1;1.2.1 Linear Delay Differential Equation;23
4.2.2;1.2.2 Numerical Simulation of DDEs;25
4.2.3;1.2.3 Nonlinear Delay Differential Equations;26
4.3;1.3 Salient Features of Chaotic Time-Delay Systems;28
4.4;References;28
5;Chapter 2 Linear Stability and Bifurcation Analysis;31
5.1;2.1 Introduction;31
5.2;2.2 Linear Stability Analysis;31
5.2.1;2.2.1 Example: Linear Delay Differential Equation;33
5.3;2.3 A Geometric Approach to Study Stability;34
5.3.1;2.3.1 Example: Linear Delay Differential Equation;35
5.4;2.4 A General Approach to Determine Linear Stability of Equilibrium Points;36
5.4.1;2.4.1 Characteristic Equation;36
5.4.2;2.4.2 Stability Conditions;36
5.4.3;2.4.3 Stability Curves/Surfaces in the (,a,b) Parameter Space;37
5.4.4;2.4.4 Extension to Coupled DDEs/Complex Scalar DDEs;38
5.4.5;2.4.5 Bifurcation Analysis;39
5.4.6;2.4.6 Results of Stability Analysis;39
5.4.7;2.4.7 A Theorem on the Stability of Equilibrium Points;40
5.4.8;2.4.8 Example: Linear Delay Differential Equation;40
5.5;References;43
6;Chapter 3 Bifurcation and Chaos in Time-Delayed Piecewise Linear Dynamical System;44
6.1;3.1 Introduction;44
6.2;3.2 Simple Scalar First Order Piecewise Linear DDE;45
6.2.1;3.2.1 Fixed Points and Linear Stability;46
6.3;3.3 Numerical Study of the Single Scalar Piecewise Linear Time-Delay System;49
6.3.1;3.3.1 Dynamics in the Pseudospace;49
6.3.2;3.3.2 Transients;50
6.3.3;3.3.3 One and Two Parameter Bifurcation Diagrams;54
6.3.4;3.3.4 Lyapunov Exponents and Hyperchaotic Regimes;56
6.4;3.4 Experimental Realization using PSPICE Simulation;57
6.5;3.5 Stability Analysis and Chaotic Dynamics of Coupled DDEs;59
6.5.1;3.5.1 Fixed Points and Linear Stability;59
6.6;3.6 Numerical Analysis of the Coupled DDE;62
6.6.1;3.6.1 Transients;63
6.6.2;3.6.2 One and Two Parameter Bifurcation Diagrams;64
6.7;References;66
7;Chapter 4 A Few Other Interesting Chaotic Delay Differential Equations;68
7.1;4.1 Introduction;68
7.2;4.2 The Mackey-Glass System: A Typical Nonlinear DDE;68
7.2.1;4.2.1 Mackey-Glass Time-Delay System;68
7.2.2;4.2.2 Fixed Points and Linear Stability Analysis;69
7.2.3;4.2.3 Time-Delay =0;70
7.2.4;4.2.4 Time-Delay >0;70
7.2.5;4.2.5 Numerical Simulation: Bifurcations and Chaos;75
7.2.6;4.2.6 Experimental Realization Using Electronic Circuit;77
7.3;4.3 Other Interesting Scalar Chaotic Time-Delay Systems;80
7.3.1;4.3.1 A Simple Chaotic Delay Differential Equation;80
7.3.2;4.3.2 Ikeda Time-Delay System;80
7.3.3;4.3.3 Scalar Time-Delay System with Polynomial Nonlinearity;82
7.3.4;4.3.4 Scalar Time-Delay System with Other Piecewise Linear Nonlinearities;83
7.3.5;4.3.5 Another Form of Scalar Time-Delay System;86
7.3.6;4.3.6 El Niño and the Delayed Action Oscillator;89
7.4;4.4 Coupled Chaotic Time-Delay Systems;91
7.4.1;4.4.1 Time-Delayed Chua's Circuit;91
7.4.2;4.4.2 Semiconductor Lasers;92
7.4.3;4.4.3 Neural Networks;94
7.5;References;95
8;Chapter 5 Implications of Delay Feedback: Amplitude Death and Other Effects;98
8.1;5.1 Introduction;98
8.2;5.2 Time-Delay Induced Amplitude Death;98
8.2.1;5.2.1 Theoretical Study: Single Oscillator;99
8.2.2;5.2.2 Experimental Study;102
8.3;5.3 Amplitude Death with Distributed Delay in Coupled Limit Cycle Oscillators;104
8.4;5.4 Amplitude Death in Coupled Chaotic Oscillators;106
8.5;5.5 Amplitude Death with Conjugate (Dissimilar) Coupling;109
8.6;5.6 Amplitude Death with Dynamic Coupling;111
8.7;5.7 Time-Delay Induced Bifurcations;114
8.8;5.8 Some Other Effects of Delay Feedback;115
8.9;References;116
9;Chapter 6 Recent Developments on Delay Feedback/Coupling: Complex Networks, Chimeras, Globally Clustered Chimeras and Synchronization;117
9.1;6.1 Introduction;117
9.2;6.2 Complex Networks;117
9.3;6.3 Chimera States in Delay Coupled Identical Oscillators;120
9.3.1;6.3.1 Discovery of Chimera States;120
9.3.2;6.3.2 Chimera States in Delay Coupled Systems;123
9.4;6.4 Chimera States in Delay Coupled Subpopulations: Globally Clustered States;125
9.5;6.5 Synchronization in Complex Networks with Delay;129
9.6;6.6 Controlling Using Time-Delay Feedback;130
9.6.1;6.6.1 Pyragas Time-Delay Feedback Control;131
9.6.2;6.6.2 Transient Behavior with Time-Delay Feedback;134
9.7;6.7 Further Developments;136
9.8;References;137
10;Chapter 7 Complete Synchronization of Chaotic Oscillations in Coupled Time-Delay Systems ;139
10.1;7.1 Introduction;139
10.2;7.2 Complete Synchronization in Coupled Time-Delay Systems;141
10.3;7.3 Stability Using Krasovskii-Lyapunov Theory;142
10.4;7.4 Numerical Confirmation;145
10.4.1;7.4.1 Case 1 ;146
10.4.2;7.4.2 Case 2;146
10.4.3;7.4.3 Case 3;147
10.4.4;7.4.4 Case 4;147
10.5;7.5 Conclusion;147
10.6;References;148
11;Chapter 8 Transition from Anticipatory to Lag Synchronization via Complete Synchronization;151
11.1;8.1 Introduction;151
11.2;8.2 Coupled System and the General Stability Condition;151
11.3;8.3 Coupled Piecewise Linear Time-Delay System and Stability Condition: Transition from Anticipatory to Lag Synchronization;153
11.3.1;8.3.1 Anticipatory Synchronization for 2 < 1;154
11.3.2;8.3.2 Complete Synchronization for 2 = 1;158
11.3.3;8.3.3 Lag Synchronization for 2 > 1;159
11.3.4;8.3.4 Inverse Synchronizations;161
11.4;8.4 Transition from Anticipatory to Lag via Complete Synchronization: Mackey-Glass and Ikeda Systems;165
11.4.1;8.4.1 Anticipatory Synchronization for 2 < 1;166
11.4.2;8.4.2 Complete Synchronization for 2 = 1;170
11.4.3;8.4.3 Lag Synchronization for 2 > 1;170
11.5;8.5 Inverse Synchronizations: Mackey-Glass and Ikeda Systems;173
11.6;References;175
12;Chapter 9 Intermittency Transition to Generalized Synchronization;177
12.1;9.1 Introduction;177
12.2;9.2 Broad Range (Slow/Delayed) Intermittency Transition to GS for Linear Error Feedback Coupling of the Form (x1(t)-x2(t));178
12.3;9.3 Stability Condition;179
12.4;9.4 Approximate (Intermittent) Generalized Synchronization;180
12.5;9.5 Characterization of IGS;183
12.6;9.6 Narrow Range (Immediate) Intermittency Transition to GS for Linear Direct Feedback Coupling of the Form x1(t);185
12.7;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
12.8;9.8 Narrow Range Intermittency Transition to GS for Nonlinear Direct Feedback Coupling of the Form f(x1(t-2));193
12.9;9.9 Intermittency Transition to Generalized Synchronization: Mackey-Glass & Ikeda Systems;197
12.9.1;9.9.1 Broad Range Intermittency Transition to GS;198
12.9.2;9.9.2 Narrow Range Intermittency Transition to GS;202
12.10;References;211
13;Chapter 10 Transition from Phase to Generalized Synchronization ;212
13.1;10.1 Introduction;212
13.2;10.2 Phase-Coherent and Non-phase-coherent Attractors;213
13.3;10.3 CPS in Chaotic Systems;214
13.4;10.4 CPS and Time-Delay Systems;216
13.5;10.5 CPS from Poincaré Surface of Section of the Transformed Attractor ;218
13.6;10.6 CPS from Recurrence Quantification Analysis;221
13.7;10.7 CPS from the Lyapunov Exponents;224
13.8;10.8 Concept of Localized Sets;225
13.9;10.9 Transition from Phase to Generalized Synchronization: Mackey-Glass & Ikeda Systems;226
13.9.1;10.9.1 CPS from Poincaré Section of the Transformed Attractor;228
13.9.2;10.9.2 CPS from Recurrence Quantification Analysis;229
13.9.3;10.9.3 CPS from the Lyapunov Exponents;230
13.9.4;10.9.4 CPS in Coupled Ikeda Systems;232
13.10;10.10 Summary;236
13.11;References;236
14;Chapter 11 DTM Induced Oscillating Synchronization;238
14.1;11.1 Introduction;238
14.2;11.2 Estimation of the Effect of Delay Time Modulation;239
14.2.1;11.2.1 Filling Factor;239
14.2.2;11.2.2 Length of Polygon Line;241
14.2.3;11.2.3 Average Mutual Information;241
14.3;11.3 Coupled System and Stability Condition in the Presence of Delay Time Modulation;244
14.4;11.4 Oscillating Synchronization;246
14.5;11.5 Intermittent Anticipatory Synchronization;249
14.6;11.6 Complete Synchronization;252
14.7;11.7 Intermittent Lag Synchronization;253
14.8;11.8 Complex Oscillating Synchronization;256
14.9;11.9 DTM Induced Oscillating Synchronization: Mackey-Glass & Ikeda Systems;256
14.9.1;11.9.1 Coupled Mackey-Glass Systems;256
14.9.2;11.9.2 Coupled Ikeda Systems;257
14.10;11.10 Summary;259
14.11;References;260
15;Chapter 12 Exact Solutions of Certain Time Delay Systems: The Car-Following Models;262
15.1;12.1 Introduction;262
15.2;12.2 The Car-Following Models;262
15.3;12.3 The Newell Model;263
15.4;12.4 The tanh Car-Following Model;266
15.5;12.5 Other Developments;268
15.6;References;269
16;Appendix A Computing Lyapunov Exponents for Time-DelaySystems;270
17;Appendix B A Brief Introduction to Synchronizationin Chaotic Dynamical Systems;274
18;Appendix C Recurrence Analysis;289
19;Appendix D Some More Examples of DDEs;302
20;Glossary;309
21;Index;317