Chapter
1.3.2 Pharmacokinetics and Pharmacodynamics (PK/PD)
1.3.3 Modeling Bone Remodeling Cycles—Integer Models
1.4 Variable Order Models—Creating Compact Biochemical Bone Remodeling Models
2 Nonlinear Fractional Order Boundary-Value Problems With Multiple Solutions
2.2 The Methods Procedures
2.2.1 Predictor Homotopy Analysis Method
2.2.2 Controlled Picard’s Method
2.2.3 The Prediction of Solutions Multiplicity
2.3.1 Mixed Convection Flows in a Vertical Channel
2.3.2 Heat Transfer Model and Diffusion-Reaction Model
2.3.3 The Bratu’s Problem
2.3.4 Three-Point Boundary Value Problem
3 Fractional Order System Forced-response Decomposition and Its Application
3.2 Notation and Preliminaries
3.3 Decomposition of the Forced Response
3.5.1 Approximation Procedure
3.7 Discussion and Extensions
4 Exact Solution of Linear Fractional Distributed Order Systems With Exponential Order Weight Functions
4.5 Exact Response of a Distributed Order RC Electrical Circuit
5 Fractional Order Time-Varying-Delay Systems: A Delay-Dependent Stability Criterion by Using Diffusive Representation
5.4 Stability Analysis in Presence of Time-Varying Delays
5.4.1 Time-Delay-Independent Stability
5.4.2 Time-Delay-Dependent Stability
5.5 Feedback Stabilization
5.5.1 Feedback Stabilization Based on Time-Delay-Independent Stability Condition
5.5.2 Feedback Stabilization Based on Time-Delay-Dependent Stability Condition
5.6.1 Approximation of the Fractional Order Derivatives and Consistent Initialization
6 Fractional Order Error Models With Parameter Constraints
6.2.1 Fractional Calculus
6.2.2 Stability of Fractional Order Systems
6.2.3 Additional Tools for the Analysis of FOS
6.3 Analysis of Fractional Order Error Model 2 With Parameter Constraints
6.3.1 Fractional Order Error Model 2
6.3.2 FOEM2 With Parameter Constraints
6.3.3 Stability Analysis of FOEM2 With Parameter Constraints
6.4 Analysis of Fractional Order Error Model 3 With Parameter Constraints
6.4.1 Fractional Order Error Model 3
6.4.2 FOEM3 With Parameter Constraints
6.4.3 Stability Analysis of FOEM3 With Parameter Constraints
6.5.1 Simulation Specifications
6.5.2 Simulation Results: Ideal Conditions
6.5.3 Simulation Results: Noisy Environment
7 Controllability of Single-valued and Multivalued Fractional Stochastic Differential Equations
7.2 Controllability Results of Single-valued and Multivalued FSDEs by Using (a,k)-Regularized Families of Bounded Linear Op...
7.2.1 Approximate Controllability of Multivalued Fractional Stochastic Integro-differential Equation
7.2.1.2 Approximate Controllability Result
7.2.2 Solvability and Optimal Control Results for FSDEs
7.2.2.2 Existence of Mild Solution
7.2.2.3 Existence of Optimal Control Result
7.3 Controllability Result of Single-valued Fractional Stochastic Differential Equation by Using Analytic Resolvent Operators
7.3.1.1 Existence of Mild Solution
7.3.1.2 Existence of Optimal Control
8 Controllability of Fractional Higher Order Stochastic Integrodifferential Inclusions
9 Design of Fractional Order Fuzzy Sliding Mode Controller for Nonlinear Complex Systems
9.2 Dynamic Model of Manipulator System
9.3 Design and Implementation of Sliding Mode Controllers
9.4 IOSMCPD Controller Design
9.5 FOFSMCPD Controller Design
9.5.1 Stability Analysis Using Lyapunov Stability Criteria
9.5.3 Fractional Order Operator Implementation
9.6 Optimization of Controller Gains by GA
9.6.2 Tuning of Controller for Trajectory Tracking Task
9.7 Detailed Performance Evaluation
9.7.1 Disturbance Rejection When Disturbance Is Injected at Controller Output
9.7.2 Disturbance Rejection When Disturbance Is Injected at Plant Output
9.7.3 Robustness Testing: Uncertainty in Length
9.8 Robustness Testing: Uncertainty in Mass
10 Sliding Mode Stabilization and Synchronization of Fractional Order Complex Chaotic and Hyperchaotic Systems
10.2.1 Fractional Order Calculus Preliminaries
10.2.2 Chaotic and Hyperchaotic Systems Used in This Study
10.3 Terminal Sliding Mode Stabilization of Fractional Order Complex Chaotic Systems
10.3.1 Terminal Sliding Mode Control Law for the Chen Chaotic System
10.3.2 Terminal Sliding Mode Control Law for the Lorenz Hyperchaotic System
10.4 Adaptive Terminal Sliding Mode Synchronization for Chaotic and Hyperchaotic Systems
10.5 Numerical Simulation Examples
10.5.1 Stabilization of Complex Chaotic and Hyperchaotic Systems
10.5.1.1 Stabilization of the Fractional Order Complex Chen Chaotic System
10.5.1.2 Stabilization of the Fractional Order Complex Lorenz Hyperchaotic System
10.5.2 Synchronization of Fractional Order Complex Chaotic and Hyperchaotic Systems
10.5.2.1 Nonidentical Synchronization of Complex Chaotic Systems
10.5.2.2 Identical Synchronization of Fractional Order Complex Hyperchaotic Systems
11 Multiswitching Synchronization of Commensurate Fractional Order Hyperchaotic Systems Via Active Control
11.3 The Review and the Approximation of a Fractional Operator
11.5 Stability of Fractional Order Systems
11.7 Multiswitching Complete Synchronization of Fractional Order Hyperchaotic Systems
12 Dual Combination Synchronization Scheme for Nonidentical Different Dimensional Fractional Order Systems Using Scaling Ma...
12.3 Application of the Scheme
12.3.1 Dual Combination Synchronization With Order n%3em
12.3.2 Dual Combination Synchronization With Order m%3en
13 On the Fractional Order Generalized Discrete Maps
13.2 Integer Order Generalized Discrete Maps
13.2.1 Maps With Signed Control Parameter
13.2.2 Maps With Scaling Parameters
13.2.2.1 Generalized Logistic Map With Scaling Parameters
13.2.2.2 Generalized Tent Map With Scaling Parameters
13.2.3 Maps With Shaping Parameters
13.2.3.1 Logistic Map With Shaping Parameters
13.2.3.2 Tent Map With Shaping Parameters
13.2.4 Maps With Both Scaling and Shaping Parameters
13.3 Fractional Order Generalized Logistic Map
13.3.1 Dynamics of the Wu Fractional Logistic Map
13.3.1.1 Fixed Points Analysis
13.3.2 Dynamics of Generalized Discrete Fractional Logistic Map
13.3.2.1 Fixed Points and Range of ρ
13.3.2.2 Stability Analysis
13.3.3 Vertical Scaling and Zooming Maps
13.3.4 Generalized Fractional Logistic Map Design
13.4 Fractional Order Generalized Tent Map
13.4.1 Effects of Main System Parameters
13.4.2 Effects of Scaling Parameters
14 Applications of Continuous-time Fractional Order Chaotic Systems
14.1 Introduction to Chaotic Systems
14.2 Simulation of Fractional Order Chaotic Systems
14.3.1 Sliding Mode Control
14.4.1 Synchronization of Fractional Order Modified Van der Pol-Duffing Circuit
14.4.2 Synchronization of Electrically Coupled Neuron Systems
14.4.3 Sliding Mode Synchronization of Uncertain Chaotic Systems
14.5 Communications and Encryption
14.5.1 Two-Channel Secure Communication
14.5.2 Substitution Boxes
14.6.1 Evaluation of Arteriovenous Stenosis in Hemodialysis Patients
14.6.2 Generalized Chaotic Susceptible Infected Recovered Epidemic Model
14.7.1 Brushless DC Motor
14.7.2 Permanent Magnet Synchronous Motor
14.8 FPGA Implementations
14.8.1 HDL Coder-based Implementation of FOCS
14.8.2 Grünwald–Letnikov Based FPGA Implementation of FOCS
15 Multiswitching Synchronization Between Chaotic Fractional Order Systems of Different Dimensions
15.3.1 Duffing Oscillator
15.4 Design of Controllers
15.5 Numerical Simulation of Results
16 Dynamics, Synchronization and Fractional Order Form of a Chaotic System With Infinite Equilibria
16.2 Model and Dynamics of the System With an Infinite Number of Equilibrium Points
16.3 Antisynchronization of Two Identical Systems Without Equilibrium
16.4 Fractional Order Form of the System With Infinite Equilibria
17 Chaotic Properties of Various Types of Hidden Attractors in Integer and Fractional Order Domains
17.2 Numerical Solution of Integer and Fractional Order Differential Equations
17.3 Survey of Some Systems With Hidden Attractors
17.4 Sensitivity to Parameter Variations
17.4.1 A Conservative Hidden Attractor With No Equilibria and Quadratic Nonlinearities: System (1)
17.4.2 A Dissipative Hidden Attractor With No Equilibria and Quadratic Nonlinearities: System (2)
17.4.3 A Dissipative Hidden Attractor With No Equilibria and Quadratic Nonlinearities: System (16)
17.4.4 A Dissipative Hidden Attractor With Single Stable Equilibrium Point: System (19)
17.5 Sensitivity to Fractional Orders
18 Parameters Identification of Fractional Order Permanent Magnet Synchronous Motor Models Using Chaotic Meta-Heuristic Alg...
18.2.1 Fractional order PMSM
18.4 Proposed Chaotic Optimization Techniques Overview
18.4.2 Chaotic Grey Wolf Optimization Algorithm
18.4.2.1 Grey Wolf Optimization Algorithm Overview (GWO)
18.4.2.2 Chaotic Grey Wolf Optimizer Strategy (CGWO)
18.4.3 Chaotic Grasshopper Optimization Algorithm
18.4.3.1 Grasshopper Optimization Algorithm Overview
18.4.3.2 Chaotic Grasshopper Optimizer Strategy (CGOA)
18.5 Simulation and Results
18.5.1 Commensurate Fractional Order Permanent Magnet Synchronous Motor Model
18.5.2 Incommensurate Fractional Order Permanent Magnet Synchronous Motor Model
19 Control and Synchronization of a Fractional Order Hyperchaotic System via Backstepping and Active Backstepping Approach
19.1.1 Introduction to Chaotic Systems
19.1.2 Introduction to Fractional Calculus
19.1.3 Introduction to Fractional Order Chaotic Systems
19.2 Preliminaries of Fractional Calculus
19.2.1 Numerical Solution of Fractional Order Differential Equations
19.2.2 Stability of Fractional Order Nonlinear Dynamical System
19.4 Control and Synchronization via Backstepping Technique
19.4.1 Controller Design for Stabilization
19.4.2 Controller Design for Synchronization
19.5 Control and Synchronization by Using Active Backstepping Technique
19.5.1 Controller Design for Stabilization
19.5.2 Synchronization Using Active Backstepping
20 Enhanced Fractional Order Indirect Fuzzy Adaptive Synchronization of Uncertain Fractional Chaotic Systems Based on the V...
20.2 Basic Definition and Preliminaries for Fractional Calculus
20.3 Basics and Description of the T–S Fuzzy Systems
20.4 Fuzzy Adaptive Robust H∞ Control: Sliding Mode Approach (VSC)
21 On the Synchronization and Recurrence of Fractional Order Chaotic Systems
21.2 General Information on Chaotic Dynamic Systems
21.2.1 Concept of the Deterministic Chaos
21.2.1.1 The Recurrence Plot
21.2.1.2 The Recurrence Quantification Analysis
21.3 Basics on Fractional Order Systems
21.3.1 Concept of Fractional Order System
21.3.1.1 Definition of Grünwald–Letnikov
21.3.1.2 Definition of Riemann–Liouville
21.3.1.3 Definition of Caputo
21.3.2 The n-Dimensional Fractional Order Chaotic System
21.3.3 The General Numerical Solution of the Fractional Differential Equation and Stability
21.4 Topology of Fractional Order Space
21.4.1 The Hausdorff–Besicovitch Dimension
21.4.2 The Correlation Dimension
21.4.3 The Maximum Lyapunov Exponent
21.5 Generalized Synchronization of Fractional Order Chaotic Systems
21.5.1 Review of the Traditional Methods of Synchronization of Fractional Order Chaotic Systems
21.5.2 The Topological Synchronization
21.6 Topological Synchronization for Some Examples of Fractional Order Chaotic Systems
21.6.1 Topological Synchronization of Fractional Order Lorenz’s Systems
21.6.1.1 The Fractional Order Lorenz’s System
21.6.1.2 Algorithm of Topological Synchronization of Fractional Order Lorenz’s Systems
21.6.2 Topological Synchronization of Fractional Order Rössler’s Systems
21.6.2.1 The Fractional Order Rössler’s System
21.6.2.2 Algorithm of Topological Synchronization of Fractional Order Rössler’s Systems
Mathematical Techniques of Fractional Order Systems
( Advances in Nonlinear Dynamics and Chaos (ANDC) )
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