Control of Complex Nonlinear Systems with Delay

by Philipp Hïvel

Overview

This research addresses delay effects in nonlinear systems, which are ubiquitous in various fields of physics, chemistry, biology, engineering, and even in social and economic systems. They may arise as a result of processing times or due to the finite propagation speed of information between the constituents of a complex system. Time delay has two complementary, counterintuitive and almost contradictory facets. On the one hand, delay is able to induce instabilities, bifurcations of periodic and more complicated orbits, multi-stability and chaotic motion. On the other hand, it can suppress instabilities, stabilize unstable stationary or periodic states and may control complex chaotic dynamics. This thesis deals with both aspects, and presents novel fundamental results on the controllability of nonlinear dynamics by time-delayed feedback, as well as applications to lasers, hybrid-mechanical systems, and coupled neural systems.    

Product Details

ISBN-13:	9783642141096
Publisher:	Springer Berlin Heidelberg
Publication date:	09/30/2010
Series:	Springer Theses
Edition description:	2011
Pages:	253
Product dimensions:	6.30(w) x 9.30(h) x 0.80(d)

Table of Contents

1 Introduction 1

References 6

2 Time-Delayed Feedback Control 11

2.1 Control Method 12

2.2 Extended Time-Delayed Feedback 15

2.3 Coupling Schemes 17

2.4 Extensions 22

2.5 Linear Stability Analysis 27

2.6 Transfer Function 29

2.7 Intermediate Conclusion 36

References 36

3 Control of Steady States 43

3.1 Model Equations 44

3.2 Time-Delayed Feedback 47

3.2.1 Unstable Focus 47

3.2.2 Saddle Point 58

3.3 Extended Time-Delayed Feedback 62

3.4 Latency Effects 68

3.4.1 Time-Delayed Feedback 68

3.4.2 Extended Time-Delayed Feedback 70

3.5 Phase-dependent Coupling 74

3.5.1 Unstable Focus 75

3.5.2 Saddle Point 79

3.5.3 Extended Time-Delayed Feedback 82

3.5.4 Two Feedback Phases 86

3.6 Asymptotic Properties for Large Delays 91

3.7 Intermediate Conclusion 100

References 101

4 Refuting the Odd Number Limitation Theorem 105

4.1 Review of the Odd Number Limitation Theorem 106

4.2 Model Equations of the Counterexample 111

4.3 Domains of Control 119

4.4 Rotating Waves and Symmetry 130

4.5 Fold Bifurcation 133

4.6 Intermediate Conclusion 144

References 145

5 Control of Neutral Delay-Differential Equations 149

5.1 Substructuring or Hybrid Testing 150

5.2 Model Equations 153

5.3 Asymptotic Properties for Large Delays 160

5.4 Control by Time-Delayed Feedback 165

5.5 Intermediate Conclusion 172

References 173

6 Neural Systems 175

6.1 Single FitzHugh-Nagumo System 176

6.2 Two Coupled FitzHugh-Nagumo Systems 186

6.3 Single FitzHugh-Nagumo System and Time-Delayed Feedback 197

6.4 Two Coupled FitzHugh-Nagumo Systems and (Extended) Time-Delayed Feedback 208

6.5 Coupling Effects of Time-Delayed Feedback 223

6.6 Towards Networks 232

6.7 Intermediate Conclusion 239

References 240

7 Summary and Outlook 245

About the Author 249

Index 251