具有分数维、时滞和同步的复杂系统 英文PDF电子书下载

1 New Treatise in Fractional Dynamics&Dumitru Baleanu 1

1.1 Introduction 1

1.2 Basic definitions and properties of fractional derivatives and integrals 3

1.3 Fractional variational principles and their applications 10

1.3.1 Fractional Euler-Lagrange equations for discrete systems 11

1.3.2 Fractional Hamiltonian formulation 13

1.3.3 Lagrangian formulation of field systems with fractional derivatives 20

1.4 Fractional optimal control formulation 23

1.4.1 Example 24

1.5 Fractional calculus in nuclear magnetic resonance 27

1.6 Fractional wavelet method and its applications in drug analysis 32

References 35

2 Realization of Fractional-Order Controllers:Analysis,Synthesis and Application to the Velocity Control of a Servo System&Ramiro S.Barbosa,Isabel S.Jesus,Manuel F.Silva,J.A.Tenreiro Machado 43

2.1 Introduction 43

2.2 Fractional-order control systems 45

2.2.1 Basic theory 45

2.2.2 Fractional-Order controllers and their implementation 47

2.3 Oustaloup's frequency approximation method 49

2.4 The experimental modular servo system 50

2.5 Mathematical modelling and identification of the servo system 50

2.6 Fractional-order real-time control system 53

2.7 Ziegler-Nichols tuning rules 54

2.7.1 Ziegler-Nichols tuning rules:quarter decay ratio 55

2.7.2 Ziegler-Nichols tuning rules:oscillatory behavior 59

2.7.3 Comments on the results 61

2.8 A simple analytical method for tuning fractional-order controllers 63

2.8.1 The proposed analytical tuning method 65

2.9 Application of optimal fractional-order controllers 69

2.9.1 Tuning of the PID and PIλD controllers 70

2.10 Conclusions 77

References 78

3 Differential-Delay Equations&Richard Rand 83

3.1 Introduction 83

3.2 Stability of equilibrium 84

3.3 Lindstedt's method 85

3.4 Hopf bifurcation formula 88

3.4.1 Example 190

3.4.2 Derivation 91

3.4.3 Example 2 92

3.4.4 Discussion 93

3.5 Transient behavior 94

3.5.1 Example 94

3.5.2 Exact solution 95

3.5.3 Two variable expansion method(also known as multiple scales) 95

3.5.4 Approach to limit cycle 97

3.6 Center manifold analysis 97

3.6.1 Appendix:The adjoint operator A 107

3.7 Application to gene expression 108

3.7.1 Stability of equilibrium 109

3.7.2 Lindstedt's method 111

3.7.3 Numerical example 113

3.8 Exercises 114

References 115

4 Analysis and Control of Deterministic and Stochastic Dynamical Systems with Time Delay&Jian-Qiao Sun,Bo Song 119

4.1 Introduction 119

4.1.1 Deterministic systems 120

4.1.2 Stochastic systems 122

4.1.3 Methods of solution 122

4.1.4 Outline of the chapter 124

4.2 Abstract Cauchy problem for DDE 124

4.2.1 Convergence with Chebyshev nodes 126

4.3 Method of semi-discretization 127

4.3.1 General time-varying systems 129

4.3.2 Feedback controls 130

4.3.3 Analysis of the method of semi-discretization 133

4.3.4 High order control 138

4.3.5 Optimal estimation 139

4.3.6 Comparison of semi-discretization and higher order control 140

4.4 Method of continuous time approximation 143

4.4.1 Control problem formulations 144

4.5 Spectral properties of the CTA method 146

4.5.1 A low-pass filter based CTA method 149

4.5.2 Example of a first order linear system 150

4.6 Stability studies of time delay systems 153

4.6.1 Stability with Lyapunov-Krasovskii functional 153

4.6.2 Stability with Padé approximation 155

4.6.3 Stability with semi-discretization 156

4.6.4 Stability of a second order LTI system 156

4.7 Control of LTI systems 163

4.8 Control of the Mathieu system 167

4.9 An experimental validation 172

4.10 Supervisory control 174

4.10.1 Supervisory Control of the LTI System 175

4.10.2 Supervisory control of the periodic system 178

4.11 Method of semi-discretization for stochastic systems 181

4.11.1 Mathematical background 181

4.11.2 Stability analysis 183

4.12 Method of finite-dimensional markov process(FDMP) 184

4.12.1 Fokker-Planck-kolmogorov(FPK)equation 185

4.12.2 Moment equations 186

4.12.3 Reliability 187

4.12.4 First-passage time probability 188

4.12.5 Pontryagin-Vittequations 189

4.13 Analysis of stochastic systems with time delay 190

4.13.1 Stability of second order stochastic systems 190

4.13.2 One Dimensional Nonlinear System 196

References 198

5 Synchronization of Dynamical Systems in Sense of Metric Functionals of Specific Constraints&Albert C.J.Luo 205

5.1 Introduction 205

5.2 System synchronization 208

5.2.1 Synchronization of slave and master systems 208

5.2.2 Generalized synchronization 214

5.2.3 Resultant dynamical systems 216

5.2.4 Metric functionals 220

5.3 Single-constraint synchronization 223

5.3.1 Synchronicity 223

5.3.2 Singularity to constraint 227

5.3.3 Synchronicity with singularity 231

5.3.4 Higher-order singularity 232

5.3.5 Synchronization to constraint 236

5.3.6 Desynchronization to constraint 252

5.3.7 Penetration to constraint 257

5.4 Multiple-constraint synchronization 261

5.4.1 Synchronicity to multiple-constraints 261

5.4.2 Singularity to constraints 264

5.4.3 Synchronicity with singularity to multiple constraints 267

5.4.4 Higher-order singularity to constraints 270

5.4.5 Synchronization to all constraints 274

5.4.6 Desynchronization to all constraints 279

5.4.7 Penetration to all constraints 284

5.4.8 Synchronization-desynchronization-penetration 287

5.5 Conclusions 294

References 294

6 The Complexity in Activity of Biological Neurons&Yong Xie,Jian-Xue Xu 299

6.1 Complicated firing patterns in biological neurons 300

6.1.1 Time series of membrane potential 300

6.1.2 Firing patterns:spiking and bursting 300

6.2 Mathematical models 306

6.2.1 HH model 306

6.2.2 FitzHugh-Nagumo model 307

6.2.3 Hindmarsh-Rose model 308

6.3 Nonlinear mechanisms of firing patterns 309

6.3.1 Dynamical mechanisms underlying TypeⅠexcitability and TypeⅡexcitability 309

6.3.2 Dynamical mechanism for the onset of firing in the HH model 310

6.3.3 TypeⅠexcitability and TypeⅡexcitability displayed in the Morris-Lecar model 311

6.3.4 Change in types of neuronal excitability via bifurcation control 314

6.3.5 Bursting and its topological classification 322

6.3.6 Bifurcation,chaos and Crisis 324

6.4 Sensitive responsiveness of aperiodic firing neurons to external stimuli 326

6.4.1 Experimental phenomena 326

6.4.2 Nonlinear mechanisms 328

6.5 Synchronization between neurons 334

6.5.1 Significance of synchronization in the nervous system 334

6.5.2 Coupling:electrical coupling and chemical coupling 335

6.6 Role of noise in the nervous system 337

6.6.1 Constructive role:stochastic resonance and coherence resonance 337

6.6.2 Stochastic resonance:When does it not occur in neuronal models? 338

6.6.3 Global dynamics and stochastic resonance of the forced FitzHugh-Nagumo neuron model 339

6.6.4 A novel dynamical mechanism of neural excitability for integer multiple spiking 342

6.6.5 A Further Insight into Stochastic Resonance in an Integrate-and-fire Neuron with Noisy Periodic Input 345

6.6.6 Signal-to-noise ratio gain of a noisy neuron that transmits subthreshold periodic spike trains 352

6.6.7 Mechanism of bifurcation-dependent coherence resonance of Morris-Lecar Model 352

6.7 Analysis of time series of interspike intervals 353

6.7.1 Return map 353

6.7.2 Phase space reconstruction 353

6.7.3 Extraction of unstable periodic orbits 355

6.7.4 Nonlinear prediction and surrogate data methods 356

6.7.5 Nonlinear characteristic numbers 358

6.8 Application 362

6.9 Conclusions 363

References 363