Solution and Characteristic Analysis of Fractional-Order Chaotic Systems

Preface
Introduction
Contents
1 Introduction
	1.1 Overview of Chaos
		1.1.1 The Origin and Development of Chaos
		1.1.2 Definition of Chaos
		1.1.3 Characteristics of Chaos
	1.2 Overview of Fractional-Order Chaotic Systems
		1.2.1 Definitions and Properties of Fractional Calculus
		1.2.2 Overview of Solving Algorithms for Fractional-Order Chaotic Systems
	1.3 Review on the Analysis Methods of Fractional-Order Chaotic Systems
		1.3.1 Study on the Dynamical Behavior of Fractional-Order Chaotic Systems
		1.3.2 Complexity of Fractional-Order Chaotic System and Its Definition
		1.3.3 Complexity Analysis of Fractional-Order Chaotic Systems
	1.4 Overview of the Application of Fractional Chaotic Systems
		1.4.1 Circuit Implementation of Fractional-Order Chaotic Systems
		1.4.2 Synchronization Control of Fractional-Order Chaotic Systems
		1.4.3 Application of Fractional-Order Chaotic Systems in Secure Communication
	References
2 Frequency-Domain Approximation Method
	2.1 Algorithm Principle
	2.2 Dynamic Simulation
	2.3 Algorithm Instance
	References
3 Predictor–Corrector Algorithm
	3.1 Algorithm Description
	3.2 Numerical Simulation
	3.3 Algorithm Example: Dynamical Analysis of the Fractional-Order Simplified Lorenz Hyperchaotic System
		3.3.1 Dynamics with the Variation of the System Order
		3.3.2 Dynamics with the Variation of the Frequency ω
	References
4 Adomian Decomposition Method
	4.1 Algorithm Description
	4.2 Algorithm Example
		4.2.1 Fractional-Order Lorenz Chaotic System
		4.2.2 Fractional-Order Simplified Lorenz Chaos System
	4.3 Adomian Decomposition Method for Fractional-Order Time-Delay Differential Equations
	References
5 Performance Comparison of Solution Algorithms
	5.1 Uniqueness of Solutions of Fractional-Order Differential Equations
	5.2 Accuracy and Performance
	5.3 Influence of Different Factors on the Minimum Order of Chaos
		5.3.1 Minimum Order of Chaos with Different Algorithms
		5.3.2 Simulation Step Size on the Minimum Order of Chaos
		5.3.3 Lowest Order of Chaos with Different Derivative Orders
		5.3.4 System Parameters on the Lowest Order of Chaos
		5.3.5 The Optimal Minimum Order of Chaos
	References
6 Dynamics of Fractional-Order Chaotic Systems
	6.1 Lyapunov Exponent Spectrum Calculation Algorithm
	6.2 Dynamical Characterization of a Family of Fractional-Order Lorenz Systems
		6.2.1 Fractional-Order Lorenz Chaotic System
		6.2.2 Fractional-Order Simplified Lorenz Chaotic System
		6.2.3 Fractional-Order Lorenz Hyperchaotic System
		6.2.4 Fractional-Order Rössler Chaotic System
		6.2.5 Fractional-Order Lorenz–Stenflo Chaotic System
		6.2.6 Fractional-Order Simplified Lorenz Hyperchaotic System
	References
7 Complexity Analysis of Fractional-Order Chaotic System
	7.1 Behavioral Complexity Algorithm
		7.1.1 Research on Multivariate Permutation Entropy Algorithm
		7.1.2 Complexity Analysis of Fractional-Order Chaotic Systems
		7.1.3 Research on Improved Multi-Scale Permutation Entropy Algorithm
		7.1.4 Complexity Analysis of Fractional-Order Chaotic System
	7.2 Structural Complexity Algorithm
		7.2.1 SE and C0 Complexity
		7.2.2 Structural Complexity Analysis of Fractional-Order Chaotic System
	References
8 Circuit Design and Realization of Fractional-Order Chaotic System
	8.1 Research on Analog Circuit of Fractional Chaos System
	8.2 Design and Implementation of Fractional-Order Chaotic System on DSP
		8.2.1 Algorithm for Solving Fractional-Order Simplified Lorenz System
		8.2.2 Hardware Design of Fractional-Order Chaotic System
		8.2.3 Software Design of Fractional Chaos System
	8.3 FPGA Design and Implementation of Fractional-Order Chaotic System
		8.3.1 Design and Optimization of Circuit Structure
		8.3.2 Design of Fractional Simplified Lorenz System Chaotic Signal Generator
		8.3.3 Design of Floating-Point Arithmetic Units
		8.3.4 Experimental Results and Analysis
	References
9 Applications of Fractional-Order Chaotic Systems in Secure Communications
	9.1 Synchronous Control of the Fractional-Order Chaotic Systems
		9.1.1 Coupled Synchronization of Fractional-Order Chaotic Systems
		9.1.2 Generalized Function Projection Synchronization for Fractional-Order Chaotic Systems
		9.1.3 Network Synchronization of Fractional-Order Chaotic Systems
	9.2 Design of the Fractional-Order Chaotic Pseudo-Random Sequence Generator
		9.2.1 Design of Chaotic Pseudo-Random Sequence Generator Based on the Fractional-Order Simplified Lorenz System
		9.2.2 Performance Tests for Chaotic Pseudo-Random Sequences
		9.2.3 Design of the Fractional-Order Lorenz Hyperchaotic Pseudo-Random Sequence Generator
	9.3 Study of Fractional-Order Chaotic Image Encryption Algorithm
		9.3.1 Image Encryption Algorithm Based on Fractional-Order Hyperchaotic System
		9.3.2 Security Analysis of Image Encryption Algorithm
	9.4 Fractional-Order Chaotic Spread Spectrum Communication Systems
		9.4.1 Spread Spectrum Codes in Spread Spectrum Communication Systems
		9.4.2 Design of a Single-User Fractional-Order Chaotic Spread Spectrum Communication System Based on Correlated Reception
		9.4.3 Performance Analysis of the Spread Spectrum Communication Systems
		9.4.4 Design of a Multi-user Chaotic Spread Spectrum Communication System Based on Correlated Reception
		9.4.5 Design of a Multi-User Chaotic Spread Spectrum Communication System Based on the Rake Reception
	References
10 Solution and Characteristic Analysis of Fractional-Order Discrete Chaotic System
	10.1 Research Progress of Fractional-Order Discrete Chaotic Systems
	10.2 Definition of Discrete Fractional-Order Difference
	10.3 Characteristic Analysis of Fractional-Order Discrete Chaotic System
		10.3.1 The Fractional-Order Logistic Map
		10.3.2 The Fractional-Order Hénon Map
		10.3.3 The Fractional-Order High-Dimensional Chaotic Map
	References
Appendix Program Code
Reference