Contents
Preface
Acknowledgment
1 Discrete Fractional Calculus
1.1. Introduction
1.2. Preliminaries
1.3. Fractional Difference Operators
1.3.1. Fractional Difference Sum Operator
1.3.1.1. Composing fractional sums and differences
1.3.1.2. Leibenitz formula
1.3.2. Riemann–Liouville Difference Operator
1.3.2.1. Composing with fractional sum and difference operators
1.3.2.2. Initial value problem
1.3.3. Caputo Fractional Difference Operator
1.3.3.1. Properties of fractional left-Caputo difference operator
1.3.3.2. Taylor difference formula
1.3.3.3. Initial value problem
1.4. Other Fractional Difference Operators
1.4.1. Grunwald–Letnikov Fractional Difference Operator
1.4.2. Fractional h-Difference Operators
1.5. Transform Methods
1.5.1. Z-Transform Method
1.5.2. Laplace Transform Method
1.6. Stability of Fractional Order Difference Systems
1.6.1. Stability of Fractional Order Linear Systems
1.6.2. Stability of Fractional Order Nonlinear Difference Systems
2 Chaotic Methods and Tests
2.1. Introduction
2.2. Discrete Chaos
2.2.1. Characterization of Chaotic Dynamical System
2.2.1.1. Sensitive initial condition
2.2.2. Strange Attractor
2.2.3. Chaotic Discrete Systems
2.3. Classical Tools to Detect Fractional Chaos
2.3.1. Bifurcation Diagrams
2.3.2. Lyapunov Exponents
2.3.2.1. Calculating Lyapunov exponents via Jacobi method
2.3.2.2. Calculating Lyapunov exponents of discrete fractional map via Jacobian matrix algorithm
2.3.2.3. Application in fractional Duffing map
2.4. 0–1 Test Method
2.4.1. Applications of the 0–1 Test on Fractional Order Maps
2.5. C0 Complexity Algorithm
2.5.1. Applying C0 Complexity for Analyzing the Complexity of Fractional Maps
2.6. Approximate Entropy
2.6.1. Applying ApEn for Analyzing the Complexity of Fractional Chaotic Duffing Map
3 Chaos in 2D Discrete Fractional Systems
3.1. Introduction
3.2. Fractional Quadratic Maps
3.2.1. Fractional-Order Hénon Map
3.2.1.1. Lyapunov exponents method
3.2.1.2. The 0–1 test method
3.2.2. Fractional Order Flow Map
3.2.3. Fractional Order Lorenz Map
3.3. Fractional Trigonometric Maps
3.3.1. Fractional Order Sine Map
3.3.1.1. Bifurcation diagrams, Lyapunov exponents and phase portraits
3.3.1.2. The 0–1 test method
3.3.2. Fractional Order Sine–Sine Map
3.3.2.1. Dynamics of the fractional order sine–sine map
3.4. Fractional Rational Maps
3.4.1. Fractional Order Rulkov Map
3.4.2. Fractional Order Chang et al. Map
3.5. Fractional Unified Maps
3.5.1. Fractional Order H´enon–Lozi Type Map
3.5.1.1. Bifurcations and largest Lyapunov exponents
3.5.1.2. The 0–1 test method
3.5.2. Fractional Order Zeraoulia–Sprott Map
4 Chaos in 3D Discrete Fractional Systems
4.1. Introduction
4.2. Fractional Generalized Hénon Map
4.3. Fractional Generalized Hénon Map with Lorenz-Like Attractors
4.3.1. Bifurcation and Chaotic Attractors
4.3.2. The 0–1 Test Method
4.4. Fractional Stefanski Map
4.5. Fractional Rössler System
4.6. Fractional Wang Map
4.7. Fractional Grassi–Miller Map
4.7.1. Dynamics Analysis on Varying α
4.7.2. Dynamics Analysis on Varying ν
4.7.3. The 0–1 Test
4.8. Fractional Cournot Game Model
4.8.1. The Fractional Order Cournot Game Model with Long Memory
4.8.2. Stability Analysis
4.8.3. Bifurcation Analysis and Numerical Simulations
4.8.4. The 0–1 Test Method
4.8.5. Approximate Entropy
5 Applications of Fractional Chaotic Maps
5.1. Control of Fractional Chaotic Maps
5.1.1. Nonlinear Control Laws
5.1.1.1. 2D fractional map
5.1.1.2. 3D fractional map
5.1.2. Linear Control Laws
5.1.2.1. 2D fractional discrete system
5.1.2.2. 3D fractional discrete system
5.2. Synchronization in Fractional Chaotic Maps
5.2.1. Generalized Synchronization (GS)
5.2.2. Inverse Generalized Synchronization (IGS)
5.2.3. Q–S Synchronization
5.2.4. The Coexistence of Different Synchronization Types
5.2.4.1. Coexistence of PS, FSHPS and GS
5.2.4.2. Coexistence of IFSHPS and IGS
5.3. Encryption Based on Fractional Discrete Chaotic Maps
5.3.1. Design of Pseudo-Random Bit Generator (PRBG)
5.3.2. Encryption of Electrophysiological Signal
5.4. Electronic Implementation of Fractional Chaotic Maps
5.4.1. The Fractional Map with the Grunwald–Letnikov Operator
5.4.2. Hardware Implementation
Bibliography
Index