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## Über dieses Buch

The book reports on the latest advances in and applications of fractional order control and synchronization of chaotic systems, explaining the concepts involved in a clear, matter-of-fact style. It consists of 30 original contributions written by eminent scientists and active researchers in the field that address theories, methods and applications in a number of research areas related to fractional order control and synchronization of chaotic systems, such as: fractional chaotic systems, hyperchaotic systems, complex systems, fractional order discrete chaotic systems, chaos control, chaos synchronization, jerk circuits, fractional chaotic systems with hidden attractors, neural network, fuzzy logic controllers, behavioral modeling, robust and adaptive control, sliding mode control, different types of synchronization, circuit realization of chaotic systems, etc.
In addition to providing readers extensive information on chaos fundamentals, fractional calculus, fractional differential equations, fractional control and stability, the book also discusses key applications of fractional order chaotic systems, as well as multidisciplinary solutions developed via control modeling. As such, it offers the perfect reference guide for graduate students, researchers and practitioners in the areas of fractional order control systems and fractional order chaotic systems.

## Inhaltsverzeichnis

### Comparative Study on Fractional Order PID and PID Controllers on Noise Suppression for Manipulator Trajectory Control

Abstract
The main contribution of this chapter is to demonstrate the sensor and controller noise suppression capabilities of the best tuned Fractional Order-Proportional plus Integral plus Derivative (FO-PID) and classical PID controllers in closed-loop. A complex non-linear and coupled system, a 2-link rigid planar manipulator was considered for the study as it encounters noise in many forms such as sensor and controller noise during the operation in industry. Uniform White Noise (UWN) and Gaussian White Noise (GWN) were considered both for the sensor and the controller in the closed-loop and a comparative study was performed for FO-PID and PID controllers. Both the controllers were tuned using Genetic Algorithm and all the simulations were performed in LabVIEW environment. The simulation results have revealed that FO-PID controller demonstrates superior sensor and controller noise suppression as compared to conventional PID controller in the closed-loop.
Vineet Kumar, K. P. S. Rana

### Control of the Temperature of a Finite Diffusive Interface Medium Using the CRONE Controller

Abstract
This chapter deals with the control of the temperature across a finite diffusive interface medium using the CRONE controller (French acronym: Commande Robuste d’Ordre Non Entier). In fact, the plant transfer function presents two special properties: a fractional integrator of order 0.5 and a delay factor of a fractional order (when controlling the temperature far from the boundary where the density of flux is applied). The novel approach of this work resides by the use of a fractional controller that would control a fractional order plant. Also note that the choice of the CRONE generation is important as this controller is developed in three generations: the first generation CRONE strategy is particularly appropriate when the desired open-loop gain crossover frequency ω u is within a frequency range where the plant frequency response is asymptotic (this frequency band will be called a plant asymptotic-behavior band). As for the second generation, it is defined when ω u is within a frequency range where the plant uncertainties are gain-like along with a constant phase variation. Concerning the third generation, it would be applied when both a gain and a phase variations are observed when dealing with plant’s uncertainties. This generation will not be treated in this chapter due to some space constraints. Thus, this chapter will present some case scenarios which will lead to the use of the first two CRONE generations when using three different plants: the first one is constituted of iron, the second of aluminum and the third of copper with variable lengths L and several placements of the temperature sensor x. Simulation results will show the temperature variation across the diffusive interface medium in both time and frequency domains using Matlab and Simulink. These results show how the temperature behaves at different positions for the three materials in use.
X. Moreau, R. Abi Zeid Daou, F. Christophy

### Grey Predictor Assisted Fuzzy and Fractional Order Fuzzy Control of a Moving Cart Inverted Pendulum

Abstract
In this chapter, a fractional order fuzzy PD controller with grey predictor (FOFPD-GP) is presented for effective control of a moving cart inverted pendulum. FOFPD-GP was tuned with the help of Genetic Algorithm for minimum settling time and its performance has been assessed using Integral of Absolute Error (IAE) and Integral of Square Error (ISE). Further, a comparative study of FOFPD-GP with its potential counterparts such as fuzzy PD with grey predictor (FPD-GP) controller, a fractional order fuzzy PD (FOFPD) controller and fuzzy PD (FPD) controller has also been carried out to assess its relative performance. Additionally, the pendulum was subjected to the impulse and sinusoidal disturbances and the disturbance rejection capabilities of the investigated controllers were analyzed and have been presented in this chapter. The simulation results revealed that FOFPD-GP controller outperformed all the other controllers under study by offering least IAE and ISE values.
Amanvir Singh Sidana, Akarsh Kumar, Akshit Kanda, Vineet Kumar, K. P. S. Rana

### Design with Fractional-Order Type Controllers

Abstract
This chapter focuses on $${{H}}_{{\infty }}$$ performance design for fractional-delay systems with fractional-order $${{PI}}^{\lambda} {{D}}^{\mu}$$ type controllers, including $${{PI}}^{\lambda} {{D}}^{\mu}$$ controller, $${{PI}}^{\lambda}$$ controller and $${{PD}}^{\mu}$$ controller. The method adopted here is based on parameter plane approach. Firstly, the stabilizing region boundary lines in the plain of the two controller’s gains are drawn for other parameters of the controller to be fixed, and the stabilizing region is identified using a graphical stability criterion applicable to fractional-delay systems. Secondly, in the stabilizing region, the modern $${{H}}_{{\infty }}$$-norm constraint of sensitivity function or complementary sensitivity function is mapped into the stabilizing region by means of the explicit algebraic equations obtained according to the definition of $${{H}}_{{\infty }}$$-norm. Thirdly, in the stabilizing region, the classical phase-margin and gain-margin curves are drawn using the technique of the gain and phase margin tester. Thus, the co-design of the modern and classical performances is realized. Finally, in time-domain, the dynamic behaviors and the robustness to the plant uncertainties are simulated via Matlab toolbox and compared with integer-order PID controller. Also, the influence of varying the fractional orders ($$\lambda$$ and $$\mu$$) on the step responses is simulated.
De-Jin Wang

### On the Electronic Realizations of Fractional-Order Phase-Lead-Lag Compensators with OpAmps and FPAAs

Abstract
It is well known that the fractional-order phase-lead-lag compensators can achieve control objectives that are not always possible by using their integer-order counterparts. However, up to now one can find only a few of publications discussing the strategies for parameters’ tuning of these compensators, with only simulation results reported. This is due in part to the implicit difficulties on the implementation of circuit elements with frequency responses of the form $$s^{\pm \lambda }$$ that are named “fractances”. In this regard, there exist approximations with rational functions, but the drawback is the difficulty to approximate the required values with the ones of the commercially-available resistances and capacitors. Consequently, fractional compensators have not been appreciated by the industry as it is in the academia. Therefore, motivated by the lack of reported implementations, this chapter is structured as a tutorial that deals with the key factors to perform, with the frequency-domain approach, the design, simulation and implementation of integer-order and fractional-order phase-lead-lag compensators. The circuit implementations are performed with Operational Amplifiers (OpAmps) and with Field Programmable Analog Arrays (FPAA). Emphasis is focused in the obtaining of commercially-available values of resistances and capacitors. Therefore, the design procedure starts with the use of equations that provide the exact and unique solution for each parameter of the compensator, avoiding conventional trial-and-error procedures. Then, five OpAmp-based configurations for integer-order and fractional-order realizations are described in terms of basic analog building blocks, such as integrators or differential amplifiers, among others. The corresponding design equations are also provided. Then, six examples are presented for both, OpAmp-based and FPAA-based implementations with the simulation and experimental results discussed regarding other results reported in the literature.
Carlos Muñiz-Montero, Luis A. Sánchez-Gaspariano, Carlos Sánchez-López, Víctor R. González-Díaz, Esteban Tlelo-Cuautle

### Robust Adaptive Supervisory Fractional Order Controller for Optimal Energy Management in Wind Turbine with Battery Storage

Abstract
To address the challenges of poor grid stability, intermittency of wind speed, lack of decision-making, and low economic benefits, many countries have set strict grid codes that wind power generators must accomplish. One of the major factors that can increase the efficiency of wind turbines (WTs) is the simultaneous control of the different parts in several operating area. A high performance controller can significantly increase the amount and quality of energy that can be captured from wind. The main problem associated with control design in wind generator is the presence of asymmetric in the dynamic model of the system, which makes a generic supervisory control scheme for the power management of WT complicated. Consequently, supervisory controller can be utilized as the main building block of a wind farm controller (offshore), which meets the grid code requirements and can increased the efficiency of WTs, the stability and intermittency problems of wind power generation. This Chapter proposes a new robust adaptive supervisory controller for the optimal management of a variable speed turbines (VST) and a battery energy storage system (BESS) in both regions (II and III) simultaneously under wind speed variation and grid demand changes. To this end, the second order sliding mode (SOSMC) with the adaptive gain super-twisting control law and fuzzy logic control (FLC) are used in the machine side, BESS side and grid side converters. The control objectives are fourfold:
(i)
Control of the rotor speed to track the optimal value;

(ii)
Maximum Power Point Tracking (MPPT) mode or power limit mode for adaptive control;

(iii)
Maintain the DC bus voltage close to its nominal value;

(iv)
Ensure: a smooth regulation of grid active and reactive power quantity, a satisfactory power factor correction and a high harmonic performance in relation to the AC source and eliminating the chattering effect.

Results of extensive simulation studies prove that the proposed supervisory control system guarantees to track reference signals with a high harmonic performance despite external disturbance uncertainties.

### Robust Adaptive Interval Type-2 Fuzzy Synchronization for a Class of Fractional Order Chaotic Systems

Abstract
This chapter presents a novel Robust Adaptive Interval Type-2 Fuzzy Logic Controller (RAIT2FLC) equipped with an adaptive algorithm to achieve synchronization performance for fractional order chaotic systems. In this work, by incorporating the $$H^{\infty }$$ tracking design technique and Lyapunov stability criterion, a new adaptive fuzzy control algorithm is proposed so that not only the stability of the adaptive type-2 fuzzy control system is guaranteed but also the influence of the approximation error and external disturbance on the tracking error can be attenuated to an arbitrarily prescribed level via the H tracking design technique. The main contribution in this work is the use of the interval type-2 fuzzy logic controller and the numerical approximation method of Grünwald-Letnikov in order to improve the control and synchronization performance comparatively to existing results. By introducing the type-2 fuzzy control design and robustness tracking approach, the synchronization error can be attenuated to a prescribed level, even in the presence of high level uncertainties and noisy training data. A simulation example on chaos synchronization of two fractional order Duffing systems is given to verify the robustness of the proposed AIT2FLC approach in the presence of uncertainties and bounded external disturbances.
Khatir Khettab, Yassine Bensafia, Samir Ladaci

### Optimal Fractional Order Proportional—Integral—Differential Controller for Inverted Pendulum with Reduced Order Linear Quadratic Regulator

Abstract
The objective of this chapter is to present an optimal Fractional Order Proportional—Integral-Differential (FOPID) controller based upon Reduced Linear Quadratic Regulator (RLQR) using Particle Swarm Optimization (PSO) algorithm and compared with PID controller. The controllers are applied to Inverted Pendulum (IP) system which is one of the most exciting problems in dynamics and control theory. The FOPID or PID controller with a feed-forward gain is responsible for stabilizing the cart position and the RLQR controller is responsible for swinging up the pendulum angle. FOPID controller is the recent advances improvement controller of a conventional classical PID controller. Fractional-order calculus deals with non-integer order systems. It is the same as the traditional calculus but with a much wider applicability. Fractional Calculus is used widely in the last two decades and applied in different fields in the control area. FOPID controller achieves great success because of its effectiveness on the dynamic of the systems. Designing FOPID controller is more flexible than the standard PID controller because they have five parameters with two parameters over the standard PID controller. The Linear Quadratic Regulator (LQR) is an important approach in the optimal control theory. The optimal LQR needs tedious tuning effort in the context of good results. Moreover, LQR has many coefficients matrices which are designer dependent. These difficulties are talked by introducing RLQR. RLQR has an advantage which allows for the optimization technique to tune fewer parameters than classical LQR controller. Moreover, all coefficients matrices that are designer dependent are reformulated to be included into the optimization process. Tuning the controllers’ gains is one of the most crucial challenges that face FOPID application. Thanks to the Metaheuristic Optimization Techniques (MOTs) which solves this dilemma. PSO technique is one of the most widely used MOTs. PSO is used for the optimal tuning of the FOPID controller and RLQR parameters. The control problem is formulated to attain the combined FOPID controllers’ gains with a feed forward gain and RLQR into a multi-dimensions control problem. The objective function is designed to be multi-objective by considering the minimum settling time, rise time, undershoot and overshoot for both the cart position and the pendulum angle. It is evident from the simulation results, the effectiveness of the proposed design approach. The obtained results are very promising. The design procedures are presented step by step. The robustness of the proposed controllers is tested for internal and external large and fast disturbances.
M. E. Mousa, M. A. Ebrahim, M. A. Moustafa Hassan

### Towards a Robust Fractional Order PID Stabilizer for Electric Power Systems

Abstract
This chapter deals with the design and application of a robust Fractional Order PID (FOPID) power system stabilizer tuned by Genetic Algorithm (GA). The system’s robustness is assured through the application of Kharitonov’s theorem to overcome the effect of system parameter’s changes within upper and lower pounds. The FOPID stabilizer has been simplified during the optimization using the Oustaloup’s approximation for fractional calculus and the “nipid” toolbox of Matlab during simulation. The objective is to keep robust stabilization with maximum attained degree of stability against system’s uncertainty. This optimization will be achieved with the proper choice of the FOPID stabilizer’s coefficients (kp, ki, kd, λ, and δ) as discussed later in this chapter. The optimization has been done using the GA which limits the boundaries of the tuned parameters within the allowable domain. The calculations have been applied to a single machine infinite bus (SMIB) power system using Matlab and Simulink. The results show superior behavior of the proposed stabilizer over the traditional PID.
Magdy A.S. Aboelela, Hisham M. Soliman

### Application of Fractional Order Controllers on Experimental and Simulation Model of Hydraulic Servo System

Abstract
Hydraulic Servo System (HSS) plays an important role in industrial applications and other fields such as plastic injection machine, material testing machines, flight simulator and landing gear system of the aircraft. The main reason of using hydraulic systems in many applications is that, it can provide a high torque and high force. The hydraulic control problems can be classified into force, position, acceleration and velocity problems. This chapter presents a study of using fractional order controllers for a simulation model and experimental position control of hydraulic servo system. It also presents an implementation of a non-linear simulation model of Hydraulic Servo System (HSS) using MATLAB/SIMULINK based on the physical laws that govern the studied system. A simulation model and experimental hardware of hydraulic servo system have been implemented to give an acceptable closed loop control system. This control system needs; for example, a conventional controller or fractional order controller to make a hydraulic system stable with acceptable steady state error. The utilized optimization techniques for tuning the proposed fractional controller are Genetic Algorithm (GA). The utilized simulation model in this chapter describes the behavior of BOSCH REXROTH of Hydraulic Servo System (HSS). Furthermore the fractional controllers and conventional controllers will be tuned by Genetic Algorithm. In addition, the hydraulic system has a nonlinear effect due to the friction between cylinders and pistons, fluid compressibility and valve dynamics. Due to these effects, the simulation and experimental results show that using fractional order controllers will give better response, minimum performance indices values, better disturbance rejection, and better sinusoidal trajectory than the conventional PID/PI controllers. It also shows that the fractional controller based on Genetic Algorithm has the desired robustness to system uncertainties such as the perturbation of the viscous friction, Coulomb friction, and supply pressure.
M. El-Sayed M. Essa, Magdy A. S. Aboelela, M. A. M. Hassan

### Control and Synchronization of Fractional-Order Chaotic Systems

Abstract
The chaotic dynamics of fractional-order systems and their applications in secure communication have gained the attention of many recent researches. Fractional-order systems provide extra degrees of freedom and control capability with integer-order differential equations as special cases. Synchronization is a necessary function in any communication system and is rather hard to be achieved for chaotic signals that are ideally aperiodic. This chapter provides a general scheme of control, switching and generalized synchronization of fractional-order chaotic systems. Several systems are used as examples for demonstrating the required mathematical analysis and simulation results validating it. The non-standard finite difference method, which is suitable for fractional-order chaotic systems, is used to solve each system and get the responses. Effect of the fractional-order parameter on the responses of the systems extended to fractional-order domain is considered. A control and switching synchronization technique is proposed that uses switching parameters to decide the role of each system as a master or slave. A generalized scheme for synchronizing a fractional-order chaotic system with another one or with a linear combination of two other fractional-order chaotic systems is presented. Static (time-independent) and dynamic (time-dependent) synchronization, which could generate multiple scaled versions of the response, are discussed.
Ahmed G. Radwan, Wafaa S. Sayed, Salwa K. Abd-El-Hafiz

### Adaptive Control of a Novel Nonlinear Double Convection Chaotic System

Abstract
This research work describes a six-term novel nonlinear double convection chaotic system with two nonlinearities. First, this work presents the 3-D dynamics of the novel nonlinear double convection chaotic system and depicts the phase portraits of the system. Our novel nonlinear double convection chaotic system is obtained by modifying the dynamics of the Rucklidge chaotic system (1992). Next, the qualitative properties of the novel chaotic system are discussed in detail. The novel chaotic system has three equilibrium points. We show that the equilibrium point at the origin is a saddle point, while the other two equilibrium points are saddle-foci. The Lyapunov exponents of the novel nonlinear double convection chaotic system are obtained as $$L_1 = 0.2089$$, $$L_2 = 0$$ and $$L_3 = -3.2123$$. The Lyapunov dimension of the novel chaotic system is obtained as $$D_{L} = 2.0650$$. Next, we present the design of adaptive feedback controller for globally stabilizing the trajectories of the novel nonlinear double convection chaotic system with unknown parameters. Furthermore, we present the design of adaptive feedback controller for achieving complete synchronization of the identical novel nonlinear double convection chaotic systems with unknown parameters. The main adaptive control results are proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this research work for the novel nonlinear double convection system.
Sundarapandian Vaidyanathan, Quanmin Zhu, Ahmad Taher Azar

### On the Terminal Full Order Sliding Mode Control of Uncertain Chaotic Systems

Abstract
Over the years, several forms of sliding mode control (SMC), such as conventional SMC, terminal SMC (TSMC) and fuzzy SMC (FSMC) have been developed to cater to the control needs of complex, non-linear and uncertain systems. However, the chattering phenomenon in conventional SMC and the singularity errors in TSMC make the application of these schemes relatively impractical. In this chapter, terminal full order SMC (TFOSMC), the recent development in this line, has been explored for efficient control of the uncertain chaotic systems. Two important chaotic systems, Genesio and Arneodo-Coullet have been considered in fractional order as well as integer order dynamics. The investigated fractional and integer order chaotic systems are controlled using fractional order TFOSMC and integer order TFOSMC, respectively and the control performance has been assessed for settling time, amount of chattering, integral absolute error (IAE) and integral time absolute error (ITAE). To gauge the relative performance of TFOSMC, a comparative study with FSMC, tuned by Cuckoo Search Algorithm for the minimum IAE and amount of chattering has also been performed using settling time, amount of chattering, IAE and ITAE performances. The intensive simulation studies presented in this chapter clearly demonstrate that the settling time, amount of chattering and steady-state tracking errors offered by TFOSMC are significantly lower than that of FSMC; therefore, making TFOSMC a superior scheme.
Anchan Saxena, Apeksha Tandon, Awadhi Saxena, K. P. S. Rana, Vineet Kumar

### Stabilization of Fractional Order Discrete Chaotic Systems

Abstract
Chaos is almost ubiquitous in field of science and engineering. The insurgent and typically unpredictable behavior exhibited by nonlinear systems is seen as chaos. In recent decades, fractional (non-integer) order chaotic systems have also been developed and their applications have invited a lot of interest of research community. Along with fractional order continuous time chaotic systems, researchers have also explored fractional order discrete chaotic systems to some extent. These systems can also be exploited for the same application for which continuous versions are used, thus providing increased flexibility and reliability. Although the mathematics of fractional discrete calculus is still in development phase, still with the help of available knowledge, research community has started giving attention to this emerging field. A number of contributions are available in literature in the area of fractional discrete calculus and its applications in control systems. One can represent linear systems using fractional difference equations in state space domain. Similarly, fractional difference equations can be used to represent nonlinear dynamical systems, especially chaotic systems. Fractional order discrete chaotic systems offer a new domain of exploration to research fraternity. As the work reported is limited, so the need arises to review and consolidate it. The analysis, control and synchronization of fractional order chaotic systems is the aim of this chapter. This chapter initially, gives a brief overview of fractional difference equations and their solution. Thereafter, the results obtained so far in this area are discussed and presented. Chaotic behavior of discrete fractional versions of the famous Logistic map and Henon map is studied first. Further, control of the same class of systems is tackled. The main contribution of the work is to present analysis of control of fractional Henon map using backstepping control which is a well-known technique to researchers in area of nonlinear control. Simulation results are obtained using MATLAB and are presented at the end to validate the results.
M. K. Shukla, B. B. Sharma

### A Three-Dimensional No-Equilibrium Chaotic System: Analysis, Synchronization and Its Fractional Order Form

Abstract
Recently, a new classification of nonlinear dynamics has been introduced by Leonov and Kuznetsov, in which two kinds of attractors are concentrated, i.e. self-excited and hidden ones. Self-excited attractor has a basin of attraction excited from unstable equilibria. So, from that point of view, most known systems, like Lorenz’s system, Rössler’s system, Chen’s system, or Sprott’s system, belong to chaotic systems with self-excited attractors. In contrast, a few unusual systems such as those with a line equilibrium, with stable equilibria, or without equilibrium, are classified into chaotic systems with hidden attractor. Studying chaotic system with hidden attractors has become an attractive research direction because hidden attractors play an important role in theoretical problems and engineering applications. This chapter presents a three-dimensional autonomous system without any equilibrium point which can generate hidden chaotic attractor. The fundamental dynamics properties of such no-equilibrium system are discovered by using phase portraits, Lyapunov exponents, bifurcation diagram, and Kaplan–Yorke dimension. Chaos synchronization of proposed systems is achieved and confirmed by numerical simulation. In addition, an electronic circuit is implemented to evaluate the theoretical model. Finally, fractional-order form of the system with no equilibrium is also investigated.
Viet-Thanh Pham, Sundarapandian Vaidyanathan, Christos K. Volos, Ahmad Taher Azar, Thang Manh Hoang, Vu Van Yem

### Comparison of Three Different Synchronization Schemes for Fractional Chaotic Systems

Abstract
The importance of synchronization schemes in natural and physical systems including communication modes has made chaotic synchronization an important tool for scientist. Synchronization of chaotic systems are usually conducted without considering the efficiency and robustness of the scheme used. In this work, performance evaluation of three different synchronization schemes: Direct Method, Open Plus Closed Loop (OPCL) and Active control is investigated. The active control technique was found to have the best stability and error convergence. Numerical simulations have been conducted to assert the effectiveness of the proposed analytical results.
S. T. Ogunjo, K. S. Ojo, I. A. Fuwape

### On New Fractional Inverse Matrix Projective Synchronization Schemes

Abstract
In this study, the problem of inverse matrix projective synchronization (IMPS) between different dimensional fractional order chaotic systems is investigated. Based on fractional order Lyapunov approach and stability theory of fractional order linear systems, new complex schemes are proposed to achieve inverse matrix projective synchronization (IMPS) between n-dimension and m-dimension fractional order chaotic systems. To validate the theoretical results and to verify the effectiveness of the proposed schemes, numerical applications and computer simulations are used.

### Fractional Inverse Generalized Chaos Synchronization Between Different Dimensional Systems

Abstract
In this chapter, new control schemes to achieve inverse generalizedsynchronization (IGS) between fractional order chaotic (hyperchaotic) systems with different dimensions are presented. Specifically, given a fractional master system with dimension n and a fractional slave system with dimension m, the proposed approach enables each master system state to be synchronized with a functional relationship of slave system states. The method, based on the fractional Lyapunov approach and stability property of integer-order linear differential systems, presents some useful features: (i) it enables synchronization to be achieved for both cases $$n<m$$ and $$n>m$$; (ii) it is rigorous, being based on theorems; (iii) it can be readily applied to any chaotic (hyperchaotic) fractional systems. Finally, the capability of the approach is illustrated by synchronization examples.

### Behavioral Modeling of Chaos-Based Applications by Using Verilog-A

Abstract
In general, a system can be defined as a collection of interconnected components that transforms a set of inputs received from its environment to a set of outputs. From an engineering point of view, chaos-based applications can be classified as a electronic system where the vast majority of the internal signals used as interconnections are electrical signals. Inputs and outputs are also provided as electrical quantities, or converted from, or to, such signals using sensors or actuators. To gain insight about the overall performance of the particular chaos-based application, the whole system must be characterized and simulated simultaneously. That is not a trivial task because the complexity of each one of the blocks that comprises the system, as well as the intrinsic complex behavior of chaotic generators. In this chapter, a modeling strategy suited to represent chaos-based applications for different control parameters of chaotic systems is presented. Based on behavioral descriptions obtained from a Hardware Description Language (HDL), called Verilog-A, two applications of chaotic systems are analyzed and designed. More specifically, a chaotic sinusoidal pulse width modulator (SPWM) which is useful to develop control algorithms for motor drivers in electric vehicles, and a chaotic pulse position modulator (CPPM) widely used in communication systems are presented as cases under analysis. Those applications are coded in Verilog-A and by using different abstraction levels, the indications of the degree of detail specified on how the function is to be implemented are obtained. Therefore, these behavioral models try to capture as much circuit functionality as possible with far less implementation details than the device-level description of the electronic circuit. Several circuit simulations applying H-Spice simulator are presented to demonstrate the usefulness of the proposed models. In this manner, behavioral modeling can be a possible solution for the successful development of robust chaos-based applications due to various types of systems that can be represented and simulated by means of an abstract model.
J. M. Munoz-Pacheco, V. R. González Díaz, L. C. Gómez-Pavón, S. Romero-Camacho, F. Sánchez-Guzmán, J. Mateo-Juárez, L. Delgado-Toral, J. A. Cocoma-Ortega, A. Luis-Ramos, P. Zaca-Morán, E. Tlelo-Cuautle

### A New Method to Synchronize Fractional Chaotic Systems with Different Dimensions

Abstract
By using two scaling function matrices, the synchronization problem of different dimensional fractional order chaotic systems in different dimensions is developed in this chapter. The controller is designed to assure that the synchronization of two different dimensional fractional order chaotic systems is achieved using the Lyapunov direct method. Numerical examples and computer simulations are used to validate numerically the proposed synchronization schemes.

### A Three-Dimensional Chaotic System with Square Equilibrium and No-Equilibrium

Abstract
Recently, Leonov and Kuznetsov have introduced a new definition “hidden attractor”. Systems with hidden attractors, especially chaotic systems, have attracted significant attention. Some examples of such systems are systems with a line equilibrium, systems without equilibrium or systems with stable equilibria etc. In some interesting new research, systems in which equilibrium points are located on different special curves are reported. This chapter introduces a three-dimensional autonomous system with a square-shaped equilibrium and without equilibrium points. Therefore, such system belongs to a class of systems with hidden attractors. The fundamental dynamics properties of such system are studied through phase portraits, Poincaré map, bifurcation diagram, and Lyapunov exponents. Anti-synchronization scheme for our systems is proposed and confirmed by the Lyapunov stability. Moreover, an electronic circuit is implemented to show the feasibility of the mathematical model. Finally, we introduce the fractional order form of such system.
Viet-Thanh Pham, Sundarapandian Vaidyanathan, Christos K. Volos, Sajad Jafari, Tomas Gotthans

### A Study on Coexistence of Different Types of Synchronization Between Different Dimensional Fractional Chaotic Systems

Abstract
In this study, robust approaches are proposed to investigate the problem of the coexistence of various types of synchronization between different dimensional fractional chaotic systems. Based on stability theory of linear fractional order systems, the co-existence of full state hybrid function projective synchronization (FSHFPS), inverse generalized synchronization (IGS), inverse full state hybrid projective synchronization (IFSHPS) and generalized synchronization (GS) is demonstrated. Using integer-order Lyapunov stability theory and fractional Lyapunov method, the co-existence of FSHFPS, inverse full state hybrid function projective synchronization (IFSHFPS), IGS and GS is also proved. Finally, numerical results are reported, with the aim to illustrate the capabilities of the novel schemes proposed herein.

### Generalized Synchronization of Different Dimensional Integer-Order and Fractional Order Chaotic Systems

Abstract
In this work different control schemes are proposed to study the problem of generalized synchronization (GS) between integer-order and fractional-order chaotic systems with different dimensions. Based on Lyapunov stability theory of integer-order differential systems, fractional Lyapunov-based approach and nonlinear controllers, different criterions are derived to achieve generalized synchronization. The effectiveness of the proposed control schemes are verified by numerical examples and computer simulations.

### A New Fractional-Order Jerk System and Its Hybrid Synchronization

Abstract
In this chapter, a new Jerk chaotic system with a piecewise nonlinear (PWNL) function and its fractional-order (FO) generalization are proposed. Both the FO and the PWNL function, serving as chaotic generators, make the proposed system more adopting for electrical engineering applications. The highly complex dynamics of the novel system are investigated by theoretical analysis pointing out its elementary characteristics such as the Lyapunov exponents, the attractor forms and the equilibrium points. To focus on the application values of the novel FO system in multilateral communication, hybrid synchronization (HS) with ring connection is investigated. For such schema, where all systems are coupled on a chain, complete synchronization (CS) and complete anti-synchronization (AS) co-exist where the state variables of the first system couple the Nth system and the state variables of the Nth system couple the $$(N-1)th$$ system. Simulations results prove that the synchronization problem is achieved with success for the multiple coupled FO systems.
Abir Lassoued, Olfa Boubaker

### An Eight-Term 3-D Novel Chaotic System with Three Quadratic Nonlinearities, Its Adaptive Feedback Control and Synchronization

Abstract
This research work describes an eight-term 3-D novel polynomial chaotic system consisting of three quadratic nonlinearities. First, this work presents the 3-D dynamics of the novel chaotic system and depicts the phase portraits of the system. Next, the qualitative properties of the novel chaotic system are discussed in detail. The novel chaotic system has four equilibrium points. We show that two equilibrium points are saddle points and the other equilibrium points are saddle-foci. The Lyapunov exponents of the novel chaotic system are obtained as $$L_1 = 0.4715, L_2 = 0$$ and $$L_3 = -2.4728$$. The Lyapunov dimension of the novel chaotic system is obtained as $$D_{L} = 2.1907$$. Next, we present the design of adaptive feedback controller for globally stabilizing the trajectories of the novel chaotic system with unknown parameters. Furthermore, we present the design of adaptive feedback controller for achieving complete synchronization of the identical novel chaotic systems with unknown parameters. The main adaptive control results are proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this research work for eight-term 3-D novel chaotic system.

### Dynamics of Fractional Order Complex Uçar System

Abstract
The fractional order delay differential equations are models with rich dynamical properties. Both fractional order and delay are useful in modelling memory and hereditary properties in the physical system. In this chapter, we have proposed a complex version of fractional order Uçar system with delay. The stability of the numerical methods for solving such equations is discussed. It is observed that a slight modification in the proposed system generates chaotic trajectories. The bifurcation and chaos is studied in the modified system. The delayed feedback method is used to control chaos in the system. Finally, the system is synchronized by using the method of projective synchronization.
Sachin Bhalekar

### Hyperchaos and Adaptive Control of a Novel Hyperchaotic System with Two Quadratic Nonlinearities

Abstract
Liu-Su-Liu chaotic system (2007) is one of the classical 3-D chaotic systems in the literature. By introducing a feedback control to the Liu-Su-Liu chaotic system,we obtain a novel hyperchaotic system in this work, which has two quadratic nonlinearities. The phase portraits of the novel hyperchaotic system are displayed and the qualitative properties of the novel hyperchaotic system are discussed. We show that the novel hyperchaotic system has a unique equilibrium point at the origin, which is unstable. The Lyapunov exponents of the novel 4-D hyperchaotic system are obtained as $$L_1 = 1.1097$$, $$L_2 = 0.1584$$, $$L_3 = 0$$ and $$L_4 = -14.1666$$. The maximal Lyapunov exponent (MLE) of the novel hyperchaotic system is obtained as $$L_1 = 1.1097$$ and Lyapunov dimension as $$D_L = 3.0895$$. Since the sum of the Lyapunov exponents of the novel hyperchaotic system is negative, it follows that the novel hyperchaotic system is dissipative. Next, we derive new results for the adaptive control design of the novel hyperchaotic system with unknown parameters. We also derive new results for the adaptive synchronization design of identical novel hyperchaotic systems with unknown parameters. The adaptive control results derived in this work for the novel hyperchaotic system are proved using Lyapunov stability theory. Numerical simulations in MATLAB are shown to validate and illustrate all the main results derived in this work.

### Chaotic Planning Paths Generators by Using Performance Surfaces

Abstract
Chaotic systems have been widely used as path planning generators in autonomous mobile robots due to the unpredictability of the generated trajectories and the coverage rate of the robots workplace. In order to obtain a chaotic mobile robot, the chaotic signals are used to generate True RNGs (TRNGs), which, as is known, exploit the nondeterministic nature of chaotic controllers. Then, the bits obtained from TRNGs can be continuously mapped to coordinates ($$x_n, y_n$$) for positioning the robot on the terrain. A frequent technique to obtain a chaotic bitstream is to sample analog chaotic signals by using thresholds. However, the performance of chaotic path planning is a function of optimal values for those levels. In this framework, several chaotic systems which are used to obtain TRNGs but by computing a quasi-optimal performance surface for the thresholds is presented. The proposed study is based on sweeping the Poincaré sections to find quasi-optimal values for thresholds where the coverage rate is higher than those obtained by using the equilibrium points as reference values. Various scenarios are evaluated. First, two scroll chaotic systems such as Chua’s circuit, saturated function, and Lorenz are used as entropy sources to obtain TRNGS by using its computed performance surface. Afterwards, n-scrolls chaotic systems are evaluated to get chaotic bitstreams with the analyzed performance surface. Another scenario is dedicated to find the performance surface of hybrid chaotic systems, which are composed by three chaotic systems where one chaotic system determines which one of the remaining chaotic signals will be used to obtain the chaotic bitstream. Additionally, TRNGs from two chaotic systems with optimized Lyapunov exponents are studied. Several numerical simulations to compute diverse metrics such as coverage rate against planned points, robot’s trajectory evolution, covered terrain, and color map are carried out to analyze the resulting TRNGs. This investigation will enable to increase several applications of TRNGs by considering the proposed performance surfaces.
C. H. Pimentel-Romero, J. M. Munoz-Pacheco, O. Felix-Beltran, L. C. Gomez-Pavon, Ch. K. Volos

### Chaotic System Modelling Using a Neural Network with Optimized Structure

Abstract
In this work, the Artificial Neural Networks (ANN) are used to model a chaotic system. A method based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II) is used to determine the best parameters of a Multilayer Perceptron (MLP) artificial neural network. Using NSGA-II, the optimal connection weights between the input layer and the hidden layer are obtained. Using NSGA-II, the connection weights between the hidden layer and the output layer are also obtained. This ensures the necessary learning to the neural network. The optimized functions by NSGA-II are the number of neurons in the hidden layer of MLP and the modelling error between the desired output and the output of the neural model. After the construction and training of the neural model, the selected model is used for the prediction of the chaotic system behaviour. This method is applied to model the chaotic system of Mackey-Glass time series prediction problem. Simulation results are presented to illustrate the proposed methodology.
Kheireddine Lamamra, Sundarapandian Vaidyanathan, Ahmad Taher Azar, Chokri Ben Salah

### A New Fractional-Order Predator-Prey System with Allee Effect

Abstract
In this chapter, a new Fractional-order (FO) predator-prey system with Allee Effect is proposed and its dynamical analysis is investigated. The two case studies of weak and strong Allee Effects are considered to bring out the consequence of such extra factors on the FO system’s dynamics. Not only it will be proven, via analytic and numerical results, that the system’s stability is governed by the type of the Allee Effect but also it will be shown that such extra factor is a destabilizing force. Finally, simulation results reveal that rich dynamic behaviors of the (FO) predator-prey model are exhibited and dependent on the order value of the FO system.