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Complex Systems - Fractionality, Time-delay and Synchronization

von: Albert C.J. Luo, Jian-Qiao Sun

Springer-Verlag, 2011

ISBN: 9783642175930 , 275 Seiten

Format: PDF

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Complex Systems - Fractionality, Time-delay and Synchronization


 

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