Complex Systems - Fractionality, Time-delay and Synchronization

Title Page

Copyright Page

Table of Contents

Contributors

Chapter 1 New Treatise in Fractional Dynamics

1.1 Introduction

1.2 Basic definitions and properties of fractional derivatives and integrals

1.3 Fractional variational principles and their applications

1.3.1 Fractional Euler-Lagrange equations,for discrete systems

1.3.2 Fractional Hamiltonian formulation

1.3.2.1 A direct method with Riemann-Liouville fractional derivatives

1.3.2.2 A direct method within Caputo fractional derivatives

1.3.2.3 Fractional Ostrogradski's formulation

1.3.2.4 Example

1.3.2.5 Fractional path integral quantization

1.3.3 Lagrangianformulation offield systems with fractional derivatives

1.3.3.1 Application 1: Fractional Dirac field

1.3.3.2 Application 2: Fractional Schrodinger equation, a Lagrangian approach

1.4 Fractional optimal control formulation

1.4.1 Example

1.5 Fractional calculus in nuclear magnetic resonance

1.6 Fractional wavelet method and its applications in drug analysis

References

Chapter 2 Realization of Fractional-Order Controllers: Analysis, Synthesis and Application to the Velocity Control of a Servo System

2.1 Introduction

2.2 Fractional-order control systems

2.2.1 Basic theory

2.2.2 Fractional-Order controllers and their implementation

2.3 Oustaloup's frequency approximation method

2.4 The experimental modular servo system

2.5 Mathematical modelling and identification of the servo system

2.6 Fractional-order real-time control system

2.7 Ziegler-Nichols tuning rules

2.7.1 Ziegler-Nichols tuning rules: quarter decay ratio

2.7.2 Ziegler-Nichols tuning rules: oscillatory behavior

2.7.3 Comments on the results

2.8 A simple analytical method for tuning fractional-order controllers

2.8.1 The proposed analytical tuning method

2.9 Application of optimal fractional-order controllers

2.9.1 Tuning ofthe PID and PI.D controllers

2.10 Conclusions

References

Chapter 3 Differential-Delay Equations

3.1 Introduction

3.2 Stability of equilibrium

3.3 IJndstedt's method

3.4 HopI'bifurcation formula

3.4.1 Example 1

3.4.2 Derivation

3.4.3 Example 2

3.4.4 Discussion

3.5 Transient behavior

3.5.1 Example

3.5.2 Exact solution

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

3.5.4 Approach to limit cycle

3.6 Center manifold analysis

3.6.1 Appendix: The adjoint operator A*

3.7 Application to gene expression

3.7.1 Stability of equilibrium

3.7.2 Lilldstedt's method

3.7.3 Numerical example

3.8 Exercises

References

Chapter 4 Analysis and Control of Deterministic and Stochastic Dynamical Systems with Time Delay

4.1 Introduction

4.1.1 Deterministic systems

4.1.2 Stochastic systems

4.1.3 Methods of solution

4.1.4 Outline of the chapter

4.2 Abstract Cauchy problem for DDE

4.2.1 Convergence with Chebyshev nodes

4.3 Method of semi-discretization

4.3.1 General time-varying systems

4.3.2 Feedback controls

4.3.2.1 Optimal feedback gains

4.3.2.2 Implication of optimal feedback gains

4.3.2.3 Tracking control

4.3.3 Analysis of the method ofsemi-discretization

4.3.3.1 Linear time-invariant second order system

4.3.3.2 Mathieu equation

4.3.4 High order control

4.3.5 Optimal estimation

4.3.6 Comparison of semi-discretization and higher order control

4.4 Method of continuous time approximation

4.4.1 Control problem formulations

4.4.1.1 Full-state feedback optimal control

4.4.1.2 Output feedback optimal control

4.4.1.3 Optimal feedback gains via mapping

4.5 Spectral properties of the CTA method

4.5.1 A low-pass filter based CTA method

4.5.2 Example of a first order linear system

4.6 Stability studies of time delay systems

4.6.1 Stability with Lyapunov-Krasovskii functional

4.6.1.1 Delay independent stability conditions

4.6.1.2 Delay dependent stability conditions

4.6.2 Stability with Pade approximation

4.6.3 Stability with semi-discretization

4.6.4 Stability of a second order LTI system

Delay independent Lyapunov stability

Delay dependent Lyapunov stability

Stability by Fade approximation

4.7 Control of LTI systems

4.8 Control of the Mathieu system

4.9 An experimental validation

4.10 Supervisory control

4.10.1 Supervisory control of the LTI system

4.10.2 Supervisory control of the periodic system

4.11 Method of semi-discretization for stochastic systems

4.11.1 Mathematical background

4.11.2 Stability analysis

4.12 Method of finite-dimensional markov process (FDMP)

4.12.1 Fokker-Planck-Kolmogorov (FPK) equation

Example of the linear system

4.12.2 Moment equations

4.12.3 Reliability

4.12.4 First-passage time probability

4.12.5 Pontryagin-Vitt equations

4.13 Analysis of stochastic systems with time delay

4.13.1 Stability of second order stochastic systems

4.13.2 One Dimensional Nonlinear System

References

Chapter 5 Synchronization of Dynamical Systems in Sense of Metric Functionals of Specific Constraints

5.1 Introduction

5.2 System synchronization

5.2.1 Synchronization of slave and master systems

5.2.2 Generalized synchronization

5.2.3 Resultant dynamical systems

5.2.4 Metric functionals

5.3 Single-constraint synchronization

5.3.1 Synchronicity

5.3.2 Singularity to constraint

5.3.3 Synchronicity with singularity

5.3.4 Higher-order singularity

5.3.5 Synchronization to constraint

5.3.6 Desynchronization to constraint

5.3.7 Penetration to constraint

5.4 Multiple-constraint synchronization

5.4.1 Synchronicity to multiple-constraints

5.4.2 Singularity to constraints

5.4.3 Synchronicity with singularity to multiple constraints

5.4.4 Higher-order singularity to constraints

5.4.5 Synchronization to all constraints

5.4.6 Desynchronization to all constraints

5.4.7 Penetration to all constraints

5.4.8 Synchronization-desynchronization-penetration

5.5 Conclusions

References

Chapter 6 The Complexity in Activity of Biological Neurons

6.1 Complicated firing patterns in biological neurons

6.1.1 Time series of membrane potential

6.1.2 Firing patterns: spiking and bursting

6.2 Mathematical models

6.2.1 HH model

6.2.2 FitzHugh-Nagumo model

6.2.3 Hindmarsh-Rose model

6.3 Nonlinear mechanisms of firing patterns

6.3.1 Dynamical mechanisms underlying Type I excitability and Type II excitability

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

6.3.3 Type I excitability and Type II excitability displayed in the Morris-Lecar model

6.3.4 Change in types of neuronal excitability via bifurcation control

6.3.5 Bursting and its topological classification

6.3.6 Bifurcation, chaos and Crisis

6.4 Sensitive responsiveness of aperiodic firing neurons to external stimuli

6.4.1 Experimental phenomena

6.4.2 Nonlinear mechanisms

6.5 Synchronization between neurons

6.5.1 Significance of synchronization in the nervous system

6.5.2 Coupling: electrical coupling and chemical coupling

6.6 Role of noise in the nervous system

6.6.1 Constructive role: stochastic resonance and coherence resonance

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

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

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

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

6.6.6 Signal-to-noise ratio gain ofa noisy neuron that transmits subthreshold periodic spike trains

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

6.7 Analysis of time series of interspike intervals

6.7.1 Return map

6.7.2 Phase space reconstruction

6.7.3 Extraction of unstable periodic orbits

6.7.4 Nonlinear prediction and surrogate data methods

6.7.5 Nonlillear characteristic numbers

6.7.5.1 Correlation dimension

6.7.5.2 Lyapunov exponent

6.7.5.3 Approximate entropy

6.7.5.4 LempeI-Ziv complexity

6.8 Application

6.9 Conclusions

References

NonIinear Physical Science