Complex Systems (eBook)

Title Page 3
Copyright Page 4
Table of Contents 7
Contributors 12
Chapter 1 New Treatise in Fractional Dynamics 13
1.1 Introduction 13
1.2 Basic definitions and properties of fractional derivatives and integrals 15
1.3 Fractional variational principles and their applications 22
1.3.1 Fractional Euler-Lagrange equations,for discrete systems 23
1.3.2 Fractional Hamiltonian formulation 25
1.3.2.1 A direct method with Riemann-Liouville fractional derivatives 25
1.3.2.2 A direct method within Caputo fractional derivatives 25
1.3.2.3 Fractional Ostrogradski's formulation 26
1.3.2.4 Example 30
1.3.2.5 Fractional path integral quantization 31
1.3.3 Lagrangianformulation offield systems with fractional derivatives 32
1.3.3.1 Application 1: Fractional Dirac field 33
1.3.3.2 Application 2: Fractional Schrodinger equation, a Lagrangian approach 34
1.4 Fractional optimal control formulation 35
1.4.1 Example 36
1.5 Fractional calculus in nuclear magnetic resonance 39
1.6 Fractional wavelet method and its applications in drug analysis 44
References 47
Chapter 2 Realization of Fractional-Order Controllers: Analysis, Synthesis and Application to the Velocity Control of a Servo System 54
2.1 Introduction 54
2.2 Fractional-order control systems 56
2.2.1 Basic theory 56
2.2.2 Fractional-Order controllers and their implementation 58
2.3 Oustaloup's frequency approximation method 60
2.4 The experimental modular servo system 61
2.5 Mathematical modelling and identification of the servo system 61
2.6 Fractional-order real-time control system 64
2.7 Ziegler-Nichols tuning rules 65
2.7.1 Ziegler-Nichols tuning rules: quarter decay ratio 66
2.7.2 Ziegler-Nichols tuning rules: oscillatory behavior 70
2.7.3 Comments on the results 72
2.8 A simple analytical method for tuning fractional-order controllers 74
2.8.1 The proposed analytical tuning method 76
2.9 Application of optimal fractional-order controllers 80
2.9.1 Tuning ofthe PID and PI.D controllers 81
2.10 Conclusions 88
References 89
Chapter 3 Differential-Delay Equations 94
3.1 Introduction 94
3.2 Stability of equilibrium 95
3.3 IJndstedt's method 96
3.4 HopI'bifurcation formula 99
3.4.1 Example 1 101
3.4.2 Derivation 102
3.4.3 Example 2 103
3.4.4 Discussion 104
3.5 Transient behavior 105
3.5.1 Example 105
3.5.2 Exact solution 106
3.5.3 Two variable expansion method (also known as multiple scales) 106
3.5.4 Approach to limit cycle 108
3.6 Center manifold analysis 108
3.6.1 Appendix: The adjoint operator A* 118
3.7 Application to gene expression 119
3.7.1 Stability of equilibrium 120
3.7.2 Lilldstedt's method 122
3.7.3 Numerical example 124
3.8 Exercises 125
References 126
Chapter 4 Analysis and Control of Deterministic and Stochastic Dynamical Systems with Time Delay 129
4.1 Introduction 129
4.1.1 Deterministic systems 130
4.1.2 Stochastic systems 132
4.1.3 Methods of solution 132
4.1.4 Outline of the chapter 134
4.2 Abstract Cauchy problem for DDE 134
4.2.1 Convergence with Chebyshev nodes 136
4.3 Method of semi-discretization 137
4.3.1 General time-varying systems 139
4.3.2 Feedback controls 140
4.3.2.1 Optimal feedback gains 140
4.3.2.2 Implication of optimal feedback gains 140
4.3.2.3 Tracking control 142
4.3.3 Analysis of the method ofsemi-discretization 143
4.3.3.1 Linear time-invariant second order system 144
4.3.3.2 Mathieu equation 146
4.3.4 High order control 148
4.3.5 Optimal estimation 149
4.3.6 Comparison of semi-discretization and higher order control 150
4.4 Method of continuous time approximation 153
4.4.1 Control problem formulations 154
4.4.1.1 Full-state feedback optimal control 154
4.4.1.2 Output feedback optimal control 155
4.4.1.3 Optimal feedback gains via mapping 155
4.5 Spectral properties of the CTA method 156
4.5.1 A low-pass filter based CTA method 159
4.5.2 Example of a first order linear system 160
4.6 Stability studies of time delay systems 163
4.6.1 Stability with Lyapunov-Krasovskii functional 163
4.6.1.1 Delay independent stability conditions 163
4.6.1.2 Delay dependent stability conditions 164
4.6.2 Stability with Pade approximation 165
4.6.3 Stability with semi-discretization 166
4.6.4 Stability of a second order LTI system 166
Delay independent Lyapunov stability 167
Delay dependent Lyapunov stability 167
Stability by Fade approximation 170
4.7 Control of LTI systems 173
4.8 Control of the Mathieu system 177
4.9 An experimental validation 182
4.10 Supervisory control 184
4.10.1 Supervisory control of the LTI system 185
4.10.2 Supervisory control of the periodic system 188
4.11 Method of semi-discretization for stochastic systems 191
4.11.1 Mathematical background 191
4.11.2 Stability analysis 193
4.12 Method of finite-dimensional markov process (FDMP) 194
4.12.1 Fokker-Planck-Kolmogorov (FPK) equation 195
Example of the linear system 195
4.12.2 Moment equations 196
4.12.3 Reliability 197
4.12.4 First-passage time probability 198
4.12.5 Pontryagin-Vitt equations 199
4.13 Analysis of stochastic systems with time delay 200
4.13.1 Stability of second order stochastic systems 200
4.13.2 One Dimensional Nonlinear System 206
References 208
Chapter 5 Synchronization of Dynamical Systems in Sense of Metric Functionals of Specific Constraints 214
5.1 Introduction 214
5.2 System synchronization 217
5.2.1 Synchronization of slave and master systems 217
5.2.2 Generalized synchronization 223
5.2.3 Resultant dynamical systems 225
5.2.4 Metric functionals 229
5.3 Single-constraint synchronization 232
5.3.1 Synchronicity 232
5.3.2 Singularity to constraint 236
5.3.3 Synchronicity with singularity 240
5.3.4 Higher-order singularity 241
5.3.5 Synchronization to constraint 245
5.3.6 Desynchronization to constraint 261
5.3.7 Penetration to constraint 266
5.4 Multiple-constraint synchronization 270
5.4.1 Synchronicity to multiple-constraints 270
5.4.2 Singularity to constraints 273
5.4.3 Synchronicity with singularity to multiple constraints 276
5.4.4 Higher-order singularity to constraints 279
5.4.5 Synchronization to all constraints 283
5.4.6 Desynchronization to all constraints 288
5.4.7 Penetration to all constraints 293
5.4.8 Synchronization-desynchronization-penetration 296
5.5 Conclusions 303
References 303
Chapter 6 The Complexity in Activity of Biological Neurons 308
6.1 Complicated firing patterns in biological neurons 309
6.1.1 Time series of membrane potential 309
6.1.2 Firing patterns: spiking and bursting 309
6.2 Mathematical models 315
6.2.1 HH model 315
6.2.2 FitzHugh-Nagumo model 316
6.2.3 Hindmarsh-Rose model 317
6.3 Nonlinear mechanisms of firing patterns 318
6.3.1 Dynamical mechanisms underlying Type I excitability and Type II excitability 318
6.3.2 Dynamical mechanism for the onset of firing in the HH model 319
6.3.3 Type I excitability and Type II excitability displayed in the Morris-Lecar model 320
6.3.4 Change in types of neuronal excitability via bifurcation control 323
6.3.5 Bursting and its topological classification 331
6.3.6 Bifurcation, chaos and Crisis 333
6.4 Sensitive responsiveness of aperiodic firing neurons to external stimuli 335
6.4.1 Experimental phenomena 335
6.4.2 Nonlinear mechanisms 337
6.5 Synchronization between neurons 343
6.5.1 Significance of synchronization in the nervous system 343
6.5.2 Coupling: electrical coupling and chemical coupling 344
6.6 Role of noise in the nervous system 346
6.6.1 Constructive role: stochastic resonance and coherence resonance 346
6.6.2 Stochastic resonance: When does it not occur in neuronal models? 347
6.6.3 Global dynamics and stochastic resonance of the forced FitzHugh-Nagumo neuron model 348
6.6.4 A novel dynamical mechanism of neural excitability for integer multiple spiking 351
6.6.5 A Further Insight into Stochastic Resonance in an Integrate-and-fire Neuron with Noisy Periodic Input 354
6.6.6 Signal-to-noise ratio gain ofa noisy neuron that transmits subthreshold periodic spike trains 361
6.6.7 Mechanism of bifurcation-dependent coherence resonance of Morris-Lecar Model 361
6.7 Analysis of time series of interspike intervals 362
6.7.1 Return map 362
6.7.2 Phase space reconstruction 362
6.7.3 Extraction of unstable periodic orbits 364
6.7.4 Nonlinear prediction and surrogate data methods 365
6.7.5 Nonlillear characteristic numbers 367
6.7.5.1 Correlation dimension 367
6.7.5.2 Lyapunov exponent 368
6.7.5.3 Approximate entropy 369
6.7.5.4 LempeI-Ziv complexity 371
6.8 Application 371
6.9 Conclusions 372
References 372
NonIinear Physical Science 380