Synchronization of Integral and Fractional Order Chaotic Systems: A Differential Algebraic and Differential Geometric Approach With Selected Applications in Real-Time

Preface
Contents
List of Figures
Symbols and Acronyms
1 Control Theory and Synchronization
	1.1 The Differential-Algebraic Point of View
	1.2 Differential-Geometric Point of View
	1.3 Synchronization of Chaotic Systems
		1.3.1 Attractors
		1.3.2 Synchronization
		1.3.3 Some Examples of Synchronization
		1.3.4 Types of Synchronization
	1.4 Complete Synchronization
	1.5 Phase Synchronization
	1.6 Lag Synchronization
	1.7 Generalized Synchronization
	1.8 Some Classical Chaotic Systems
		1.8.1 Lorenz System
		1.8.2 Rössler System
		1.8.3 Chua System
		1.8.4 Colpitts System
		1.8.5 Rikitake System
		1.8.6 Duffing System
		1.8.7 Van der Pol System
	1.9 Fractional-Order Systems
		1.9.1 Fractional-Order Operator Block in Simulink
	1.10 Why Fractional Order?
	1.11 Fractional Circuit
	References
2 A  Model-Free-Based Proportional Reduced-Order Observer Design for the Synchronization of  Lorenz Systems
	2.1 Introduction
	2.2 Synchronization of Lorenz System
		2.2.1 Algebraic Observability Condition
		2.2.2 Observer Synthesis
	2.3 Numerical Results
	2.4 Concluding Remarks
	References
3 A  Model-Free Sliding Observer to the Synchronization Problem Using Geometric Techniques
	3.1 Introduction
	3.2  Observer Canonical Form of a Nonlinear System
	3.3 Sliding-Mode Observer to the Synchronization Problem
	3.4 Model-Based Observers to the Synchronization Problem
		3.4.1 Bestle–Zeitz Observer for the SynchronizationProblem
		3.4.2 Thau Observer for Synchronization
	3.5 Two Synchronization Problems
		3.5.1 Lorenz System
			3.5.1.1  Sliding-Mode Observer to the Synchronization of the Lorenz System
			3.5.1.2 Bestle–Zeitz Observer to Synchronization of Lorenz System
			3.5.1.3 Thau Observer to Synchronization of Lorenz System
		3.5.2 Chua\'s Circuit
	3.6 Concluding Remarks
	References
4  Experimental Synchronization by Means of Observers
	4.1 Introduction
	4.2 Exponential Polynomial Observer
		4.2.1 Problem Statement
	4.3 Asymptotic Reduced-Order Observer
	4.4 High-Gain Observer
	4.5 Synchronization by Means of Observers
		4.5.1 Experimental Results
		4.5.2 Synchronization of the Colpitts Oscillator Employing an Exponential Polynomial Observer
	4.6 Synchronization with a High-Gain Observer
		4.6.1 Synchronization of the Colpitts Oscillator by Means of an Asymptotic Reduced-Order Observer
		4.6.2 Rössler System
		4.6.3 Rikitake Oscillator
	4.7  Bounded Error Observer Based Design of Synchronizing Chaotic Systems
	4.8 Numerical Results
	4.9 Conclusion
	References
5 Synchronization of an Uncertain Rikitake System with Parametric Estimation
	5.1 Introduction
	5.2 Problem Statement
		5.2.1  Rikitake Model System
		5.2.2 Some Algebraic Properties and Problem Formulation
	5.3 Lyapunov-Based Formulation
	5.4 Numerical Results
	5.5 Concluding Remarks
	References
6 Secure Communications and Synchronization via a Sliding-Mode Observer
	6.1 Introduction
	6.2 Chaotic Communication Based on  a Sliding-Mode Observer
	6.3 Numerical Simulation
	6.4 Conclusions
	References
7 Synchronization and Antisynchronization of Chaotic Systems: A Differential and Algebraic Approach
	7.1 Introduction
	7.2 Statement of the Problem
		7.2.1  Antisynchronization
	7.3 Synchronization and Antisynchronization of a ColpittsOscillator
		7.3.1 Observer Design
	7.4 Numerical Results
	7.5 Concluding Remarks
	References
8 Synchronization of Chaotic Liouvillian Systems: An Application to Chua\'s Oscillator
	8.1 Introduction
	8.2 Definitions
	8.3 Problem Formulation and Main Result
		8.3.1 Observer Convergence Analysis
	8.4 Numerical and Experimental Results
		8.4.1 Numerical Results
		8.4.2 Experimental Results
	8.5 Concluding Remarks
	References
9 Synchronization of Partially Unknown Nonlinear Fractional-Order Systems
	9.1 Introduction
	9.2 On Fractional Derivatives
		9.2.1 Mittag-Leffler-Type Function
	9.3 Main Result
	9.4 Numerical Example
	9.5 Conclusions
	References
10 Generalized Synchronization via the Differential PrimitiveElement
	10.1 Introduction
	10.2 Statement of the Problem and Main Results
	10.3 Numerical Example
		10.3.1 Stability Analysis
		10.3.2 Simulation Results
	10.4 Concluding Remarks
	References
11 Generalized Synchronization for a Class of Nondifferentially Flat and Liouvillian Chaotic Systems
	11.1 Introduction
	11.2 Definitions
	11.3 Problem Statement and Methodology to Generalized Synchronization (GS)
	11.4 Generalized Synchronization of Rössler and Chua Systems
	11.5 Concluding Remarks
	References
12 Generalized Multisynchronization by Means of a Family of Dynamical Feedbacks
	12.1 Introduction
	12.2 Problem Formulation and Main Results
	12.3 Generalized Synchronization of  Multiple Decoupled Systems
	12.4 Concluding Remarks
	References
13  Fractional Generalized Synchronization in Nonlinear Fractional-Order Systems via  Dynamical Feedback
	13.1 Introduction
	13.2 Main Result
	13.3 Fractional Generalized Synchronization Between Chua and Rössler Systems
	13.4 Concluding Remarks
	References
14 An Observer for a Class of Incommensurate Fractional-Order Systems
	14.1 Introduction
	14.2 Basic Concepts
		14.2.1 Mittag-Leffler-Type Functions
	14.3 Problem Statement and Main Result
	14.4 Numerical Results
	14.5 Conclusions
	References
Appendix
	Estimation Error Convergence to the Zone  Asymptotically (Ultimately Bounded)
Index