Delay Differential Equations (eBook)

Preface 5
Contents 8
Contributors 14
Lyapunov–Krasovskii Functional Approach for Coupled Differential-Difference Equations with Multiple Delays 16
1.1 Introduction 16
1.2 Coupled Differential-Functional Equations 20
1.3 Stability of Continuous Time Difference Equations 25
1.4 Linear Coupled Differential-Difference Equations 28
1.5 Discussion and Examples 38
1.6 Concluding Remarks 42
References 43
Networked Control and Observation for Master–Slave Systems 46
2.1 Introduction 46
2.2 Exponential Stability of a Remote System Controlled Through Internet 48
2.3 Architecture of the Global Control System 53
2.4 Performance Enhancement by a Gain Scheduling Strategy 60
2.5 Conclusion 66
References 67
Developments in Control of Time-Delay Systems for Automotive Powertrain Applications 69
3.1 Introduction 69
3.2 Idle Speed Control 70
3.3 Air-to-Fuel Ratio Control 77
3.4 Observer Design for a Diesel Engine Model with Time Delay 92
3.5 Concluding Remarks 102
Appendix: Robust and Stochastic Stability 103
References 104
Stability Analysis and Control of Linear Periodic Delayed Systems Using Chebyshev and Temporal Finite Element Methods 107
4.1 Introduction 107
4.2 Stability of Autonomous and Periodic DDEs 110
4.3 Temporal Finite Element Analysis 111
4.4 Chebyshev Polynomial Expansion and Collocation 118
4.5 Application to Milling Stability 127
4.6 Control of Periodic Systems with Delay using Chebyshev Polynomials 130
4.7 Discussion of Chebyshev and TFEA Approaches 137
References 140
Systems with Periodic Coefficients and Periodically Varying Delays: Semidiscretization-Based Stability Analysis 144
5.1 Introduction 144
5.2 Stability Analysis of Systems with Periodically Varying Delays 146
5.3 Approximation of the Monodromy Matrix by using the Semidiscretization Method 148
5.4 Applications 151
5.5 Closure 164
References 165
Bifurcations, Center Manifolds, and Periodic Solutions 167
6.1 Background 167
6.2 Decomposing Ordinary Differential Equations Using Adjoints 172
6.3 An Example Application in Ordinary Differential Equations 176
6.4 Delay Differential Equations as Operator Equations 179
6.5 A Machine Tool DDE Example: Part 1 188
6.6 Computing the Bifurcated Periodic Solution on the Center Manifold 198
6.7 A Machine Tool DDE Example: Part 2 202
6.8 Simulation Results 208
References 213
Center Manifold Analysis of the Delayed Li´enard Equation 215
7.1 Introduction 215
7.2 Linear Stability Analysis 216
7.3 Operator Differential Equation Formulation 218
7.4 Center Manifold Reduction 221
7.5 Hopf Bifurcation Analysis 224
7.6 Numerical Results 225
7.7 Hopf Bifurcation in the Sunflower Equation 226
7.8 Concluding Remarks 230
References 230
Calculating Center Manifolds for Delay Differential Equations Using MapleTM 232
8.1 Introduction 232
8.2 Theory 233
8.3 Application 241
8.4 Discussion 251
References 253
Numerical Solution of Delay Differential Equations 256
9.1 Introduction 256
9.2 DDEs are not ODEs 258
9.3 Numerical Methods and Software Issues 263
9.4 Examples 269
9.5 Conclusion 280
9.6 Further Reading 280
References 280
Effects of Time Delay on Synchronization and Firing Patterns in Coupled Neuronal Systems 283
10.1 Introduction 283
10.2 Basic Concepts 286
10.3 Synchronization and Firing Patterns in Electrically Coupled Neurons with Time Delay 290
10.4 Synchronization and Firing Patterns in Coupled Neurons with Delayed Inhibitory Synapses 293
10.5 Synchronization and Firing Patterns in Coupled Neurons with Delayed Excitatory Synapses 299
10.6 Delay Effect on Multistability and Spatiotemporal Dynamics of Coupled Neuronal Activity [14] 302
10.7 Closure 307
Appendix 10.1 The Hodgkin–Huxley (HH) Model 309
Appendix 10.2 The Morris–Lecar(ML) Model 310
Appendix 10.3 The Chay Model 311
Appendix 10.4 The Fast-Spiking (FS) Model 312
References 313
Delayed RandomWalks: Investigating the Interplay Between Delay and Noise 314
11.1 Introduction 314
11.2 Simple RandomWalk 316
11.3 RandomWalks on a Quadratic Potential 325
11.4 Delayed RandomWalks 329
11.5 Postural Sway 335
11.6 Concluding Remarks 341
References 341
Index 345