Optimization of Integer/Fractional Order Chaotic Systems by Metaheuristics and their Electronic Realization

Cover
Title Page
Copyright Page
Dedication
Preface
Table of Contents
1. Numerical Methods
	1.1 Introduction
	1.2 Numerical methods for ODEs of integer order
		1.2.1 One-step methods
		1.2.2 Multi-step methods
	1.3 Stability of the numerical methods
	1.4 Fractional-order chaotic systems
	1.5 Definitions of fractional order derivatives and integrals
		1.5.1 Grünwald-Letnikov fractional integrals and derivatives
		1.5.2 Riemann-Liouville fractional integrals and derivatives
		1.5.3 Caputo fractional derivatives
	1.6 Time domain methods for fractional-order chaotic oscillators
	1.7 Simulation of the fractional-order derivative 0Dqty(t) = x(t)
		1.7.1 Approximation by applying Grünwald-Letnikov method
		1.7.2 Approximation by applying FDE12 predictor-corrector method
		1.7.3 Approximation by applying Adams-Bashforth-Moulton method
		1.7.4 Numerical simulation of 0Dt0.5y(t) = x(t) and 0Dt0.9y(t) = x(t)
2. Integer-Order Chaotic/Hyper-chaotic Oscillators
	2.1 Introduction
	2.2 Self-excited chaotic oscillators
	2.3 Amplitude scaling for the analog implementation of autonomous chaotic oscillators
		2.3.1 Amplitude scaling for the amplifier-based implementation of integer-order chaotic systems
		2.3.2 Amplitude scaling for the FPAA-based implementation of integer-order chaotic systems
	2.4 Simulating chaotic oscillators in Anadigm Designer tool
	2.5 Experimental results of the FPAA-based implementation of Lorenz system
3. Fractional-Order Chaotic/Hyper-chaotic Oscillators
	3.1 Numerical simulation in the frequency domain
		3.1.1 Approximating a fractional-order integrator by a ladder fractance topology
		3.1.2 Approximating a fractional-order integrator by a tree fractance topology
	3.2 Simulation and implementation of fractional-order chaotic oscillators by using first-order active filters
		3.2.1 Approximation of 1/sq by using first-order active filters
		3.2.2 Implementation of a fractional-order hyper-chaotic oscillator
		3.2.3 Implementation of a fractional-order multi-scroll chaotic oscillator
		3.2.4 FPAA-based implementation of the fractional-order multi-scroll chaotic oscillator
	3.3 Simulation of self-excited fractional-order chaotic oscillators by Gründwald-Letnikov method
	3.4 Simulation of self-excited multi-scroll and hyper-chaotic fractional-order chaotic oscillators by Grünwald-Letnikov method
	3.5 Simulation of hidden fractional-order chaotic oscillators by Grünwald-Letnikov method
	3.6 Simulation of fractional-order chaotic oscillators by applying Adams Bashforth-Moulton definition
4. Single-Objective Optimization Algorithms
	4.1 An optimization problem
	4.2 The Newton’s method
		4.2.1 One example
		4.2.2 Derivatives for optimization problems with several variables
	4.3 The two points gradient descent method
	4.4 The genetic algorithm
		4.4.1 One example
	4.5 Differential evolution
		4.5.1 One example
	4.6 PSO and MOL algorithms
		4.6.1 One example
	4.7 Problems with constraints
		4.7.1 The penalty function
		4.7.2 Handling constraints in the selection step
		4.7.3 One example
	4.8 LSHADE
	4.9 Optimization of DKY of integer-order chaotic oscillators by DE and PSO
5. Multi-Objective Optimization Algorithms
	5.1 Formulation of a multi-objective optimization problem
	5.2 Pareto dominance
		5.2.1 Calculating the non-dominated solutions
	5.3 NSGA-II
		5.3.1 Genetic operators
		5.3.2 Ranking and crowding distance
		5.3.3 Constraints handling
	5.4 MOEA-D
		5.4.1 Simplex-lattice design
	5.5 MOMBI-II
	5.6 Indicators
	5.7 Examples
	5.8 Optimization of the fractional-order Lorenz chaotic oscillator by applying NSGA-II
6. Single-Objective Optimization of Fractional-Order Chaotic/Hyper-chaotic Oscillators and their FPAA-based Implementation
	6.1 Fractional-Order Chaotic/Hyper-chaotic Oscillators
	6.2 Lyapunov exponents and DKY of fractional-order chaotic/hyper-chaotic oscillators
	6.3 Optimizing DKY by DE and PSO
		6.3.1 Differential Evolution Algorithm
		6.3.2 Particle Swarm Optimization Algorithm
	6.4 Search spaces, design variables and optimization results
	6.5 Implementation of optimized fractional-order chaotic/hyper-chaotic oscillators by using amplifiers
	6.6 Implementation of optimized fractional-order chaotic/hyper-chaotic oscillators by FPAA
7. Multi-Objective Optimization of Fractional-Order Chaotic/Hyper-chaotic Oscillators and their FPGA-based Implementation
	7.1 Multi-objective optimization of fractional-order chaotic/hyper-chaotic oscillators
	7.2 FPGA-based implementation of optimized fractional-order chaotic oscillators from Grünwald-Letnikov method
		7.2.1 Grünwald-Letnikov method for solving fractional-order chaotic oscillators
		7.2.2 Block diagram description of the Grünwald-Letnikov method for FPGA implementation
	7.3 VHDL descriptions of the Adder, Subtractor, Multiplier, SCM, Shift Register, and Accumulator
	7.4 Results of the FPGA-based implementation of FOCOs
8. Applications of Optimized Fractional-Order Chaotic/Hyper-chaotic Oscillators
	8.1 Synchronization of FOCOs by applying Hamiltonian forms and observer approach
		8.1.1 Master-slave synchronization of two fractional-order Chen chaotic oscillators
		8.1.2 Master-slave synchronization of two fractional-order Zhang hyper-chaotic oscillators
	8.2 Synchronization of FOCOs by applying the OPCL technique
		8.2.1 Master-slave synchronization of two hidden fractional-order Zhang hyper-chaotic oscillators
	8.3 Image encryption by using optimized FOCOs
		8.3.1 Image encryption by using hidden fractional-order Zhang hyper-chaotic oscillators
		8.3.2 FPGA-based design of a chaotic secure communication system using FOCOs
	8.4 Design of random number generators from the optimized FOCOs
References
Index
Authors Biographies