main-content

## Über dieses Buch

The book reports on the latest advances in and applications of chaos theory and intelligent control. Written by eminent scientists and active researchers and using a clear, matter-of-fact style, it covers advanced theories, methods, and applications in a variety of research areas, and explains key concepts in modeling, analysis, and control of chaotic and hyperchaotic systems. Topics include fractional chaotic systems, chaos control, chaos synchronization, memristors, jerk circuits, chaotic systems with hidden attractors, mechanical and biological chaos, and circuit realization of chaotic systems. The book further covers fuzzy logic controllers, evolutionary algorithms, swarm intelligence, and petri nets among other topics. Not only does it provide the readers with chaos fundamentals and intelligent control-based algorithms; it also discusses key applications of chaos as well as multidisciplinary solutions developed via intelligent control. The book is a timely and comprehensive reference guide for graduate students, researchers, and practitioners in the areas of chaos theory and intelligent control.

## Inhaltsverzeichnis

### A Novel Design Approach of a Nonlinear Resistor Based on a Memristor Emulator

In this chapter, a novel design method of a nonlinear resistor of type-N consisting of a memristor emulator circuit in parallel with a negative resistor, is presented. The proposed emulator is built with second-generation current conveyors (CCII) and passive elements and its pinched hysteresis loop is holding up to 20 kHz. As an example of using the designed nonlinear resistor, the simple non-autonomous Lacy circuit is chosen. The numerical as well as the simulation results, by using SPICE, reveal the richness of circuit’s dynamical behavior confirming the usefulness of the specific design method. The increased complexity that the nonlinear resistor with memristor gives to the circuit is a consequence of the effect of both signals frequency and amplitude to the memristors pinched hysteresis loop. Furthermore, the ease of the specific design method makes this proposal a very attractive option for the design of nonlinear resistors.

Ch.K. Volos, S. Vaidyanathan, V.-T. Pham, J. O. Maaita, A. Giakoumis, I. M. Kyprianidis, I. N. Stouboulos

### Dynamics, Synchronization and SPICE Implementation of a Memristive System with Hidden Hyperchaotic Attractor

The realization of memristor in nanoscale size has received considerate attention recently because memristor can be applied in different potential areas such as spiking neural network, high-speed computing, synapses of biological systems, flexible circuits, nonvolatile memory, artificial intelligence, modeling of complex systems or low power devices and sensing. Interestingly, memristor has been used as a nonlinear element to generate chaos in memristive system. In this chapter, a new memristive system is proposed. The fundamental dynamics properties of such memristive system are discovered through equilibria, Lyapunov exponents, and Kaplan–York dimension. Especially, hidden attractor and hyperchaos can be observed in this new system. Moreover, synchronization for such system is studied and simulation results are presented showing the accuracy of the introduced synchronization scheme. An electronic circuit modelling such hyperchaotic memristive system is also reported to verify its feasibility.

Viet-Thanh Pham, Sundarapandian Vaidyanathan, Christos K. Volos, Thang Manh Hoang, Vu Van Yem

### Synchronization of Fractional Chaotic and Hyperchaotic Systems Using an Extended Active Control

An extended active control technique is used to synchronize fractional order chaotic and hyperchaotic systems with and without delay. The coupling strength is set to the value less than one to achieve the complete synchronization more easily. Explicit formula for the error matrix is also proposed in this chapter. Numerical examples are given for the fractional order chaotic Liu system, hyperchaotic new system and Ucar delay system. The effect of fractional order and coupling strength on the synchronization time is studied for non-delayed cases. It is observed that the synchronization time decreases with increase in fractional order as well as with increase in coupling strength for the Liu system. For the new system, the synchronization time decreases with increase in fractional order as well as with decrease in coupling strength.

Sachin Bhalekar

### A Novel 4-D Hyperchaotic Thermal Convection System and Its Adaptive Control

In this work, we announce an eleven-term novel 4-D hyperchaotic thermal convection system with two quadratic nonlinearities. The phase portraits of the novel hyperchaotic system are depicted and the qualitative properties of the novel hyperchaotic system are discussed. The novel 4-D hyperchaotic thermal convection system is obtained by introducing a feedback control to the 3-D thermal convection system obtained by Wang et al. (J Fluid Mech 237:479–498, 1992). The Lyapunov exponents of the novel hyperchaotic thermal convection system are obtained as $$L_1 = 0.40546$$, $$L_2 = 0.03583$$, $$L_3 = 0$$ and $$L_4 = -6.44038$$. Since there are two positive Lyapunov exponents for the novel 4-D thermal convection system, it is hyperchaotic. The Maximal Lyapunov Exponent (MLE) of the novel hyperchaotic system is found as $$L_1 = 0.40546$$. Also, the Kaplan-Yorke dimension of the novel hyperchaotic system is derived as $$D_{KY} = 3.0685$$. Since the sum of the Lyapunov exponents is negative, the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the novel hyperchaotic thermal convection system with unknown parameters. Finally, an adaptive controller is also designed to achieve global chaos synchronization of the identical novel hyperchaotic thermal convection systems with unknown parameters. MATLAB simulations are depicted to illustrate all the main results derived in this work.

Sundarapandian Vaidyanathan

### Synchronization of Chaotic Dynamical Systems in Discrete-Time

In this study, we investigate the problem of chaos synchronization in discrete-time dynamical systems with different structures and diverse types. Based on Lyapunov stability theory, stability of lineare systems and nonlinear control methods some synchronization criterions are presented in 2D, 3D and N-dimensional discrete-time chaotic systems. Numerical examples and computer simulations are used to show the effectiveness and the feasibility of the proposed synchronization schemes.

Adel Ouannas, M. Mossa Al-sawalha

### Mathematical Modelling of Chaotic Jerk Circuit and Its Application in Secure Communication System

In chaos-based secure communication schemes, a message signal is modulated to the chaotic signal at transmitter and at receiver the masking signals regenerated and subtracted from the receiver signal. In order to show some interesting phenomena of three dimensional autonomous ordinary differential equations, the chaotic behavior as a function of a variable control parameter, has been studied. The initial study in this chapter is to analyze the eigenvalue structures, various attractors, bifurcation diagram, Lyapunov exponent spectrum, FFT analysis, Poincare maps, while the analysis of the synchronization in the case of bidirectional coupling between two identical generated chaotic systems, has been presented. Moreover, some appropriate comparisons are made to contrast some of the existing results. Finally, the effectiveness of the bidirectional coupling method scheme between two identical Jerk circuits in a secure communication system is presented in details.

Aceng Sambas, Mada Sanjaya WS, Mustafa Mamat, Rizki Putra Prastio

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

In this research work, we describe an eight-term 3-D novel chaotic system with three quadratic nonlinearities. First, this work describes the dynamic analysis of the novel chaotic system. The Lyapunov exponents of the eight-term novel chaotic system are obtained as $$L_1 = 4.0359, L_2 = 0$$ and $$L_3 = -29.1071$$. The Kaplan-Yorke dimension of the novel chaotic system is obtained as $$D_{KY} = 2.1384$$. Next, this work describes the adaptive feedback control of the novel chaotic system with unknown parameters. Also, this work describes the adaptive feedback synchronization of the identical novel chaotic systems with unknown parameters. The adaptive feedback control and synchronization results are proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results for the eight-term 3-D novel chaotic system.

Sundarapandian Vaidyanathan, Ahmad Taher Azar

### Qualitative Study and Adaptive Control of a Novel 4-D Hyperchaotic System with Three Quadratic Nonlinearities

In this work, we announce an eleven-term novel 4-D hyperchaotic system with three quadratic nonlinearities. The phase portraits of the eleven-term novel hyperchaotic system are depicted and the qualitative properties of the novel hyperchaotic system are discussed. The novel hyperchaotic system has a unique equilibrium at the origin, which is a saddle point. The Lyapunov exponents of the novel hyperchaotic system are obtained as $$L_1 = 2.0836$$, $$L_2 = 0.1707$$, $$L_3 = 0$$ and $$L_4 = -26.6499$$. The maximal Lyapunov exponent of the novel hyperchaotic system is found as $$L_1 = 2.0836$$. Also, the Kaplan-Yorke dimension of the novel hyperchaotic system is derived as $$D_{KY} = 3.0846$$. Since the sum of the Lyapunov exponents is negative, the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the novel hyperchaotic system with unknown parameters. Finally, an adaptive controller is also designed to achieve global chaos synchronization of the identical hyperchaotic systems with unknown parameters. MATLAB simulations are depicted to illustrate all the main results derived in this work.

Sundarapandian Vaidyanathan, Ahmad Taher Azar

### A Novel 4-D Four-Wing Chaotic System with Four Quadratic Nonlinearities and Its Synchronization via Adaptive Control Method

In this research work, we describe a ten-term novel 4-D four-wing chaotic system with four quadratic nonlinearities. First, this work describes the qualitative analysis of the novel 4-D four-wing chaotic system. We show that the novel four-wing chaotic system has a unique equilibrium point at the origin, which is a saddle-point. Thus, origin is an unstable equilibrium of the novel chaotic system. We also show that the novel four-wing chaotic system has a rotation symmetry about the $$x_3$$ axis. Thus, it follows that every non-trivial trajectory of the novel four-wing chaotic system must have a twin trajectory. The Lyapunov exponents of the novel 4-D four-wing chaotic system are obtained as $$L_1 = 5.6253$$, $$L_2 = 0$$, $$L_3 = -5.4212$$ and $$L_4 = -53.0373$$. Thus, the maximal Lyapunov exponent of the novel four-wing chaotic system is obtained as $$L_1 = 5.6253$$. The large value of $$L_1$$ indicates that the novel four-wing system is highly chaotic. Since the sum of the Lyapunov exponents of the novel chaotic system is negative, it follows that the novel chaotic system is dissipative. Also, the Kaplan-Yorke dimension of the novel four-wing chaotic system is obtained as $$D_{KY} = 3.0038$$. Finally, this work describes the adaptive synchronization of the identical novel 4-D four-wing chaotic systems with unknown parameters. The adaptive synchronization result is proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results for the novel 4-D four-wing chaotic system.

Sundarapandian Vaidyanathan, Ahmad Taher Azar

### Adaptive Control and Synchronization of Halvorsen Circulant Chaotic Systems

In this research work, we describe Halvorsen circulant chaotic systems and its qualitative properties. We show that Halvorsen circulant chaotic system is dissipative and that it has an unstable equilibrium at the origin. The Lyapunov exponents of Halvorsen circulant chaotic system are obtained as $$L_1 =0.8109$$, $$L_2 = 0$$ and $$L_3 = -4.6255$$. The Kaplan-Yorke dimension of the Halvorsen circulant chaotic system is obtained as $$D_{KY} = 2.1753$$. Next, this work describes the adaptive control of the Halvorsen circulant chaotic system with unknown parameters. Also, this work describes the adaptive synchronization of the identical Halvorsen circulant chaotic systems with unknown parameters. The adaptive feedback control and synchronization results are proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results for the Halvorsen circulant chaotic system.

Sundarapandian Vaidyanathan, Ahmad Taher Azar

### Adaptive Backstepping Control and Synchronization of a Novel 3-D Jerk System with an Exponential Nonlinearity

In this research work, we announce a seven-term novel 3-D jerk chaotic system with an exponential nonlinearity. First, we discuss the qualitative properties of the novel jerk chaotic system. The novel jerk chaotic system has a unique equilibrium point, which is a saddle-focus. Thus, the unique equilibrium point is unstable. We obtain the Lyapunov exponents of the novel jerk chaotic system as $$L_1 = 0.1066$$, $$L_2 = 0$$ and $$L_3 = -1.1047$$. Also, the Kaplan-Yorke dimension of the novel jerk chaotic system is obtained as $$D_{KY} = 2.0965$$. Next, an adaptive backstepping controller is designed to stabilize the novel jerk chaotic system with unknown system parameters. Moreover, an adaptive backstepping controller is designed to achieve complete chaos synchronization of the identical novel jerk chaotic systems with unknown system parameters. The main control results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results developed in this work.

Sundarapandian Vaidyanathan, Ahmad Taher Azar

### Generalized Projective Synchronization of a Novel Hyperchaotic Four-Wing System via Adaptive Control Method

In this research work, we announce a novel 4-D hyperchaotic four-wing system with three quadratic nonlinearities. First, this work describes the qualitative analysis of the novel 4-D hyperchaotic four-wing system. We show that the novel hyperchaotic four-wing system has a unique equilibrium point at the origin, which is a saddle-point. Thus, origin is an unstable equilibrium of the novel hyperchaotic system. The Lyapunov exponents of the novel hyperchaotic four-wing system are obtained as $$L_1 = 2.5266$$, $$L_2 = 0.1053$$, $$L_3 = 0$$ and $$L_4 = -43.0194$$. Thus, the maximal Lyapunov exponent (MLE) of the novel hyperchaotic four-wing system is obtained as $$L_1 = 2.5266$$. Since the sum of the Lyapunov exponents of the novel hyperchaotic system is negative, it follows that the novel hyperchaotic system is dissipative. Also, the Kaplan-Yorke dimension of the novel four-wing chaotic system is obtained as $$D_{KY} = 3.0612$$. Finally, this work describes the generalized projective synchronization (GPS) of the identical novel hyperchaotic four-wing systems with unknown parasmeters. The GPS is a general type of synchronization, which generalizes known types of synchronization such as complete synchronization, anti-synchronization, hybrid synchronization, etc. The main GPS result via adaptive control method is proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results for the novel 4-D hyperchaotic four-wing system.

Sundarapandian Vaidyanathan, Ahmad Taher Azar

### Hyperchaos, Control, Synchronization and Circuit Simulation of a Novel 4-D Hyperchaotic System with Three Quadratic Nonlinearities

In this work, we announce an eleven-term novel 4-D hyperchaotic system with three quadratic nonlinearities. The novel 4-D hyperchaotic system has been derived by adding a feedback control to the seven term 3-D Lu-Xiao chaotic system [1]. The phase portraits of the eleven-term novel hyperchaotic system are depicted and the qualitative properties of the novel hyperchaotic system are discussed. The novel hyperchaotic system has a unique equilibrium at the origin, which is a saddle point. Thus, the origin is an unstable equilibrium of the novel hyperchaotic system. The Lyapunov exponents of the novel hyperchaotic system are obtained as $$L_1 = 1.6023, L_2 = 0.1123, L_3 = 0$$ and $$L_4 = -22.6467$$. Also, the Kaplan-Yorke dimension of the novel hyperchaotic system is obtained as $$D_{KY} = 3.0757$$. Since the sum of the Lyapunov exponents is negative, the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the novel hyperchaotic system with unknown parameters. Moreover, an adaptive controller is also designed to achieve global chaos synchronization of the identical hyperchaotic systems with unknown parameters. Finally, an electronic circuit realization of the novel 4-D hyperchaotic system using SPICE is described in detail to confirm the feasibility of the theoretical model.

Sundarapandian Vaidyanathan, Christos K. Volos, Viet-Thanh Pham

### Complete Synchronization of Hyperchaotic Systems via Novel Sliding Mode Control

Chaos in nonlinear dynamics occurs widely in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. Synchronization of chaotic systems is an important research problem in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. In this work, we derive a novel sliding mode control method for the complete synchronization of identical chaotic or hyperchaotic systems. The general result is derived using novel sliding mode control method. The general result is established using Lyapunov stability theory. As an application of the general result, the problem of complete synchronization of identical hyperchaotic Vaidyanathan systems (2014) is studied and a new sliding mode controller is derived. The Lyapunov exponents of the hyperchaotic Vaidyanathan system are obtained as $$L_1 = 1.4252$$, $$L_2 = 0.2445$$, $$L_3 = 0$$ and $$L_4 = -17.6549$$. Since the Vaidyanathan hyperjerk system has two positive Lyapunov exponents, it is hyperchaotic. Also, the Kaplan-Yorke dimension of the Vaidyanathan hyperjerk system is obtained as $$D_{KY} = 3.0946$$. Numerical simulations using MATLAB have been shown to depict the phase portraits of the hyperchaotic Vaidyanathan system and the sliding mode controller design for the anti-synchronization of identical hyperchaotic Vaidyanathan systems.

Sundarapandian Vaidyanathan, Sivaperumal Sampath

### A Novel 3-D Conservative Jerk Chaotic System with Two Quadratic Nonlinearities and Its Adaptive Control

In this research work, we announce a six-term novel 3-D conservative jerk chaotic system with two quadratic nonlinearities. The novel conservative jerk chaotic system is obtained by adding a quadratic nonlinearity to Sprott’s 3-D conservative jerk chaotic system (1997). In this work, we first discuss the qualitative properties of the novel 3-D conservative jerk chaotic system. Conservative chaotic systems are characterized by the property that they are volume conserving. The novel conservative jerk chaotic system has two saddle-foci equilibrium points. Thus, both equilibrium points of the novel conservative jerk chaotic system are unstable. We obtain the Lyapunov exponents of the novel conservative jerk chaotic system as $$L_1 = 0.0452$$L1=0.0452, $$L_2 = 0$$L2=0 and $$L_3 = -0.0452$$L3=-0.0452. Also, the Kaplan-Yorke dimension of the conservative novel jerk chaotic system is obtained as $$D_{KY} = 3$$DKY=3. The high value of the Kaplan-Yorke dimension indicates the complexity of the novel conservative jerk chaotic system. Next, an adaptive backstepping controller is designed to stabilize the novel conservative jerk chaotic system with unknown system parameters. Moreover, an adaptive backstepping controller is designed to achieve complete chaos synchronization of the identical novel conservative jerk chaotic systems with unknown system parameters. The main control results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results on the novel 3-D conservative jerk chaotic system.

Sundarapandian Vaidyanathan

### A Novel 3-D Circulant Highly Chaotic System with Labyrinth Chaos

In this work, we describe a novel 3-D circulant highly chaotic system with labyrinth chaos. The novel chaotic system is a nine-term polynomial system with six sinusoidal nonlinearities. The phase portraits of the novel circulant chaotic system are illustrated and the dynamic properties of the novel circulant chaotic system are discussed. The novel circulant chaotic system has infinitely many equilibrium points and it exhibits labyrinth chaos. We show that all the equilibrium points of the novel circulant chaotic system are saddle-foci and hence they are unstable. The Lyapunov exponents of the novel circulant chaotic system are obtained as $$L_1 = 10.3755$$, $$L_2 = 0$$ and $$L_3 = -10.4113$$. Thus, the Maximal Lyapunov Exponent (MLE) of the novel chaotic system is obtained as $$L_1 = 10.3755$$, which is a large value. This shows that the novel 3-D circulant chaotic system is highly chaotic. Also, the Kaplan-Yorke dimension of the novel circulant highly chaotic system is obtained as $$D_{KY} = 2.9966$$. Since the Kaplan-Yorke dimension of the the novel circulant chaotic system has a large value and close to three, the novel circulant chaotic system with labyrinth chaos exhibits highly complex behaviour. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Next, we derive new results for the global chaos control of the novel circulant highly chaotic system with unknown parameters using adaptive control method. We also derive new results for the global chaos synchronization of the identical novel circulant highly chaotic systems with unknown parameters using adaptive control method. The main control results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate the phase portraits of the novel circulant highly chaotic system and also the adaptive control results derived in this work.

Sundarapandian Vaidyanathan

### Dynamic Analysis, Adaptive Control and Synchronization of a Novel Highly Chaotic System with Four Quadratic Nonlinearities

In this work, we describe an eight-term novel highly chaotic system with four quadratic nonlinearities. The phase portraits of the novel highly chaotic system are illustrated and the dynamic properties of the highly chaotic system are discussed. The novel highly chaotic system has three unstable 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 novel highly chaotic system has rotation symmetry about the $$x_3$$ axis. The Lyapunov exponents of the novel highly chaotic system are obtained as $$L_1 = 11.0572$$, $$L_2 = 0$$ and $$L_3 = -28.0494$$, while the Kaplan-Yorke dimension of the novel chaotic system is obtained as $$D_{KY} = 2.3942$$. Since the Maximal Lyapunov Exponent (MLE) of the novel chaotic system has a large value, viz. $$L_1 = 11.0572$$, the novel chaotic system is highly chaotic. Since the sum of the Lyapunov exponents is negative, the novel highly chaotic system is dissipative. Next, we derive new results for the global chaos control of the novel highly chaotic system with unknown parameters via adaptive control method. We also derive new results for the global chaos synchronization of the identical novel highly chaotic systems with unknown parameters via adaptive control method. The main adaptive control results are established using Lyapunov stability theory. MATLAB simulations are shown to depict the phase portraits of the novel highly chaotic system and also the adaptive control results derived in this work.

Sundarapandian Vaidyanathan

### Analysis, Adaptive Control and Synchronization of a Novel 3-D Chaotic System with a Quartic Nonlinearity and Two Quadratic Nonlinearities

In this work, we announce a seven-term novel 3-D chaotic system with a quartic nonlinearity and two quadratic nonlinearities. The proposed chaotic system is highly chaotic and it has interesting qualitative properties. The phase portraits of the novel chaotic system are illustrated and the dynamic properties of the highly chaotic system are discussed. The novel 3-D chaotic system has three unstable 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 novel 3-D chaotic system has rotation symmetry about the $$x_3$$ axis, which shows that every non-trivial trajectory of the system must have a twin trajectory. The Lyapunov exponents of the novel 3-D chaotic system are obtained as $$L_1 = 8.6606$$, $$L_2 = 0$$ and $$L_3 = -26.6523$$, while the Kaplan-Yorke dimension of the novel chaotic system is obtained as $$D_{KY} = 2.3249$$. Since the Maximal Lyapunov Exponent (MLE) of the novel chaotic system has a large value, viz. $$L_1 = 8.6606$$, the novel chaotic system is highly chaotic. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Next, we apply adaptive control method to derive new results for the global chaos control of the novel chaotic system with unknown parameters. We also apply adaptive control method to derive new results for the global chaos synchronization of the identical novel chaotic systems with unknown parameters. The main adaptive control results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results derived in this work.

Sundarapandian Vaidyanathan

### Qualitative Analysis and Properties of a Novel 4-D Hyperchaotic System with Two Quadratic Nonlinearities and Its Adaptive Control

A hyperchaotic attractor is typically defined as chaotic behavior with at least two positive Lyapunov exponents. Combined with one null Lyapunov exponent along the flow and one negative Lyapunov exponent to ensure the boundedness of the solution, the minimal dimension for an autonomous continuous-time hyperchaotic system is four. In this work, we announce an eleven-term novel 4-D hyperchaotic system with only two quadratic nonlinearities. The phase portraits of the eleven-term novel hyperchaotic system are depicted and the dynamic properties of the novel hyperchaotic system are discussed. We establish that the novel hyperchaotic system has three unstable equilibrium points. The Lyapunov exponents of the novel hyperchaotic system are obtained as $$L_1 = 2.5112, L_2 = 0.3327, L_3 = 0$$ and $$L_4 = -24.7976$$. The maximal Lyapunov exponent of the novel hyperchaotic system is found as $$L_1 = 2.5112$$. Also, the Kaplan-Yorke dimension of the novel hyperchaotic system is derived as $$D_{KY} = 3.1147$$. Since the sum of the four Lyapunov exponents is negative, the novel 4-D hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the novel hyperchaotic system with unknown parameters. Finally, an adaptive controller is also designed to achieve complete synchronization of the identical novel hyperchaotic systems with unknown parameters. The main adaptive control results for stabilization and synchronization of the novel hyperchaotic system are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results derived in this work for the novel 4-D hyperchaotic system.

Sundarapandian Vaidyanathan

### Global Chaos Synchronization of a Novel 3-D Chaotic System with Two Quadratic Nonlinearities via Active and Adaptive Control

In this research work, we announce a six-term novel 3-D dissipative chaotic system with two quadratic nonlinearities. First, this work describes the dynamic equations and qualitative properties of the novel chaotic system. We show that the novel chaotic system has three unstable equilibrium points. We also show that the novel chaotic system has a rotation symmetry about the $$x_3$$ axis. The Lyapunov exponents of the novel chaotic system are obtained as $$L_1 = 1.2334, L_2 = 0$$ and $$L_3 = -4.7329$$. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Also, the Kaplan-Yorke dimension of the novel chaotic system is derived as $$D_{KY} = 2.2606$$. Next, this work describes the active synchronization of identical novel chaotic systems with known parameters. Furthermore, this work describes the adaptive synchronization of identical novel chaotic systems with unknown parameters. Both the active and adaptive synchronization results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this work for the six-term novel 3-D novel chaotic system.

Sundarapandian Vaidyanathan

### A Novel 2-D Chaotic Enzymes-Substrates Reaction System and Its Adaptive Backstepping Control

In this research work, we announce a novel 2-D chaotic enzymes-substrates reaction system and discuss its adaptive backstepping control. First, this work describes the dynamic equations and qualitative properties of the novel 2-D biological chaotic system. Our novel chaotic system is obtained by modifying the equations of the 2-D enzymes-substrates reaction system with ferroelectric behaviour in brain waves obtained by Kadji et al. (Chaos Solitons Fractals 32:862–882, 2001, [27]). The Maximal Lyapunov Exponent (MLE) of the novel 2-D chaotic enzymes-substrates reaction system is obtained as $$L_1 = 0.14425$$. Next, this work describes the adaptive control of the novel 2-D chaotic enzymes-substrates reaction system via backstepping control method. Furthermore, this work describes the adaptive synchronization of identical novel 2-D chaotic enzymes-substrates reaction systems via backstepping control method. The main stabilization and synchronization results derived in this work are established via Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this work for the novel 2-D chaotic enzymes-substrates reaction system.

Sundarapandian Vaidyanathan

### Analysis, Control and Synchronization of a Novel 4-D Highly Hyperchaotic System with Hidden Attractors

In this work, we announce a ten-term novel 4-D highly hyperchaotic system with three quadratic nonlinearities. The phase portraits of the ten-term novel highly hyperchaotic system are depicted and the qualitative properties of the novel highly hyperchaotic system are discussed. We shall show that the novel hyperchaotic system does not have any equilibrium point. Hence, the novel 4-D hyperchaotic system exhibits hidden attractors. The Lyapunov exponents of the novel hyperchaotic system are obtained as $$L_1 = 13.67837$$, $$L_2 = 0.04058$$, $$L_3 = 0$$ and $$L_4 = -45.64661$$. The Maximal Lyapunov Exponent (MLE) of the novel hyperchaotic system is found as $$L_1 = 13.67837$$, which is large. Thus, the novel 4-D hyperchaotic system proposed in this work is highly hyperchaotic. Also, the Kaplan-Yorke dimension of the novel hyperchaotic system is derived as $$D_{KY} = 3.30055$$. Since the sum of the Lyapunov exponents is negative, the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the novel highly hyperchaotic system with unknown parameters. Finally, an adaptive controller is also designed to achieve global chaos synchronization of the identical novel highly hyperchaotic systems with unknown parameters. MATLAB simulations are depicted to illustrate all the main results derived in this work.

Sundarapandian Vaidyanathan

### A Novel Double Convection Chaotic System, Its Analysis, Adaptive Control and Synchronization

In this research work, we announce a novel 3-D double convection chaotic system and discuss its qualitative properties, adaptive control and synchronization. First, this work describes the dynamic equations and qualitative properties of the novel 3-D double convection chaotic system. We show that the novel 3-D double convection chaotic system has three unstable equilibrium points of which one equilibrium point is a saddle-point and the other two equilibrium points are saddle-foci. Our novel chaotic system is obtained by modifying the equations of the Rucklidge chaotic system (1992) for nonlinear double convection. The Lyapunov exponents of the novel 3-D double convection chaotic system are obtained as $$L_1 = 1.03405, L_2 = 0$$ and $$L_3 = -4.03938$$. Also, the Kaplan-Yorke dimension of the novel 3-D double convection chaotic system is derived as $$D_{KY} = 2.2560$$. Next, this work describes the global stabilization of the novel 3-D double convection chaotic system with unknown parameters via adaptive control method. Furthermore, this work describes the global chaos synchronization of identical novel 3-D double convection chaotic systems via adaptive control method. Our adaptive global stabilization and synchronization results are established via Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this work for the novel 3-D double convection chaotic system.

Sundarapandian Vaidyanathan

### A Seven-Term Novel 3-D Jerk Chaotic System with Two Quadratic Nonlinearities and Its Adaptive Backstepping Control

In this research work, we announce a seven-term novel 3-D jerk chaotic system with two quadratic nonlinearities. First, we discuss the qualitative properties of the novel 3-D jerk chaotic system. We show that the novel jerk chaotic system has two unstable equilibrium points on the $$x_1$$-axis. We establish that the novel jerk chaotic system is dissipative. Next, we obtain the Lyapunov exponents of the novel jerk chaotic system as $$L_1 = 0.11184$$, $$L_2 = 0$$ and $$L_3 = -0.61241$$. Also, we derive the Kaplan-Yorke dimension of the novel jerk chaotic system as $$D_{KY} = 2.18262$$. Next, we design an adaptive backstepping controller to stabilize the novel jerk chaotic system with unknown system parameters. We also design an adaptive backstepping controller to achieve complete chaos synchronization of the identical novel jerk chaotic systems with unknown system parameters. The main control results are established using Lyapunov stability theory. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict feedback systems. MATLAB simulations are shown to illustrate all the main results developed in this work.

Sundarapandian Vaidyanathan

### Evidence of Chaos in EEG Signals: An Application to BCI

The recent science and technology studies in neuroscience and machine learning have focused attention on investigating the functioning of the brain through nonlinear analysis. The brain is a nonlinear dynamic system, imparting randomness and nonlinearity in the EEG signals. The stochastic nature of the brain seeks the paramount importance of understanding the underlying neurophysiology. The nonlinear analysis of the dynamic structure may help to reveal the complex behavior of the brain signals. EEG signal analysis is helpful in various clinical applications to characterize the normal and diseased brain states. The EEG is used in predicting epileptic seizures, classifying the sleep stages, measuring the depth of anesthesia, and detecting the abnormal brain states. With the onset of EEG-based brain-computer interfaces, the characteristics of brain signals are used to control the devices through different mental states. Hence, the need to understand the brain state is important and crucial. In this chapter, the author introduces the theory and methods of chaos theory measurements and its applications in EEG signal analysis. A broad perspective of the techniques and implementation of the Correlation Dimension, Lyapunov Exponents, Fractal Dimension, Approximate Entropy, Sample Entropy, Hurst Exponent, Lempel-Ziv complexity, Hopf Bifurcation Theorem and Higher-order spectra is explained and their usage in EEG signal analysis is mentioned. We suggest that chaos theory provides not only potentially valuable diagnostic information but also a deeper understanding of neuropathological mechanisms underlying the brain in ways that are not possible by conventional linear analysis.

Kusuma Mohanchandra, Snehanshu Saha, K. Srikanta Murthy

### Circuit Realization of the Synchronization of Two Chaotic Oscillators with Optimized Maximum Lyapunov Exponent

The modeling, simulation and circuit realization of the synchronization of two optimized multi-scroll chaotic oscillators is described herein. The case of study is the master-slave synchronization of two multi-scroll chaotic oscillators generating four to seven scrolls, based on saturated function series. The maximum Lyapunov exponent (MLE) of the chaotic oscillator is optimized by applying meta-heuristics. We show the behavior on the synchronization for chaotic oscillators with low and high MLEs, while the synchronization is performed by generalized Hamiltonian forms and observer approach from nonlinear control theory. Numerical simulation results are given for the chaotic oscillators with and without optimized MLEs, and for their master-slave synchronization. Finally, we show the good agreement between theoretical results, SPICE simulations and the experimental results when the whole synchronized system is implemented with commercially available operational amplifiers.

V. H. Carbajal-Gómez, E. Tlelo-Cuautle, F. V. Fernández

### Evolutionary Computational Technique in Automatic Generation Control of Multi-area Power Systems with Nonlinearity and Energy Storage Unit

In this study, a new meta-heuristic based evolutionary computational technique is reported for solving Automatic Generation Control (AGC) or Load Frequency Control (LFC) issue in multi-area power system with nonlinearity and an energy storage unit. Multi-area power system consists of two area equal reheat thermal power systems with Governor Dead Band (GDB) and Generation Rate Constraint (GRC) nonlinearity and boiler dynamics and energy storage element. During normal operating conditions, there no change in system parameters (Frequency and tie-line power flow) and stability. When sudden load demand occurs in any one of interconnected power, it affects system parameters and stability and system yield damping oscillation in their response with steady state error and settling time. In order to mitigate this biggest pose the proper selection of the controller is a major issue. In power system Automatic Voltage Regulator (AVR) loop is a primary control loop and in addition Proportional-Integral-Derivative (PID) controller is proposed as a secondary controller in AGC. The better performance of power system depends on proper selection of controller gain and also depends on the selected objective function for optimization of controller gain values. A new meta-heuristic based Ant Colony Optimization (ACO) evolutionary computational technique is used for tuning of PID controller with different operating conditions. Three different objective functions Integral Square Error (ISE), Integral Time Absolute Error (ITAE) and Integral Absolute Error (IAE) are used in ACO for tuning of controller gain. An electromechanical oscillation of power system is effectively damp out by introducing an energy storage unit in two area interconnected power system because of their inherent energy storage capacity with kinetic energy of the rotor. In this study Hydrogen generative Aqua Electroliser (HAE) with a fuel cell is incorporated into the investigated power system. The response of the proposed approach with different cost functions are obtained and compared with and without considering the effect of energy storage unit in LFC problem.

K. Jagatheesan, B. Anand, K. Baskaran, Nilanjan Dey

### Fuzzy Adaptive Synchronization of Uncertain Fractional-Order Chaotic Systems

In this chapter, a fuzzy adaptive controller for a class of fractional-order chaotic systems with uncertain dynamics and external disturbances is proposed to realize a practical projective synchronization. The adaptive fuzzy systems are used to online approximate unknown system nonlinearities. The proposed control law, which is derived based on a Lyapunov approach, is continuous and ensures the stability of the closed-loop system and the convergence of the underlying synchronization errors to a neighborhood of zero. Finally, a simulation example is provided to verify the effectiveness of the proposed synchronization method.

Abdesselem Boulkroune, Amel Bouzeriba, Toufik Bouden, Ahmad Taher Azar

### Fuzzy Control-Based Function Synchronization of Unknown Chaotic Systems with Dead-Zone Input

This chapter deals with adaptive fuzzy control-based function vector synchronization between two chaotic systems with both unknown dynamic disturbances and input nonlinearities (dead-zone and sector nonlinearities). This synchronization scheme can be considered as a natural generalization of many existing projective synchronization systems (namely the function projective synchronization, the modified projective synchronization, the projective synchronization and so on). To effectively deal with the input nonlinearities, the control system is designed in a variable-structure framework. In order to approximate uncertain nonlinear functions, the adaptive fuzzy systems are incorporated in this control system. A Lyapunov approach is used to prove the boundedness of all signals as well as the exponential convergence of the corresponding synchronization errors to an adjustable region. The synchronization between two chaotic satellite systems is taken as an illustrative example to show the effectiveness of the proposed synchronization method.

Abdesselem Boulkroune, Sarah Hamel, Ahmad Taher Azar, Sundarapandian Vaidyanathan

### Feature Selection and Recognition of Muzzle Point Image Pattern of Cattle by Using Hybrid Chaos BFO and PSO Algorithms

Recognition of cattle based on muzzle point image pattern (nose print) is a well study problem in the field of animal biometrics, computer vision, pattern recognition and various application domains. Missed cattle, false insurance claims and relocation at slaughter houses are major problems throughout the world. Muzzle pattern of cattle is a suitable biometric trait to recognize them by extracted features from muzzle pattern by using computer vision and pattern recognition approaches. It is similar to human’s fingerprint recognition. However, the accuracy of animal biometric recognition systems is affected due to problems of low illumination condition, pose and recognition of animal at given distance. Feature selection is known to be a critical step in the design of pattern recognition and classifier for several reasons. It selects a discriminant feature vector set or pre-specified number of features from muzzle pattern database that leads to the best possible performance of the entire classifier in muzzle recognition of cattle. This book chapter presents a novel method of feature selection by using Hybrid Chaos Particle Swarm Optimization (PSO) and Bacterial Foraging Optimization (BFO) techniques. It has two parts: first, two types of chaotic mappings are introduced in different phase of hybrid algorithms which preserve the diversity of population and improve the global searching capability; (2) this book chapter exploited holistic feature approaches: Principal Component Analysis (PCA), Local Discriminant Analysis (LDA) and Discrete Cosine Transform (DCT) [28, 85] extract feature from the muzzle pattern images of cattle. Then, feature (eigenvector), fisher face and DCT feature vector are selected by applying hybrid PSO and BFO metaheuristic approach; it quickly find out the subspace of feature that is most beneficial to classification and recognition of muzzle pattern of cattle. This chapter provides with the stepping stone for future researches to unveil how swarm intelligence algorithms can solve the complex optimization problems and feature selection with helps to improve the cattle identification accuracy.

Santosh Kumar, Sanjay Kumar Singh

### Control of Complex Systems Using Self Organizing Fuzzy Controller

Robotic manipulators are complex multi-input multiple output systems finding lots of application in industries. Controlling such a complex system always has been an area of research owing to the inherent nonlinearities. In this work, a comparative study of Fuzzy Proportional Integral Derivative (FPID) and Self Organizing Fuzzy Controller (SOFC), applied for trajectory tracking and disturbance rejection to a two link planar rigid robotic manipulator with end-effector has been presented. Two layers of fuzzy logic controller (FLC) have been used to design SOFC in which second layer was used for adaptive mechanism and Takagi-Sugeno-Kang method has been used for inference mechanism in both the control schemes. Genetic algorithm (GA) has been used to optimize the gains of FPID and SOFC controllers for minimum Integral of Absolute Error (IAE) and Integral of Absolute Change in Controller Output. Simulation results revealed that SOFC outperformed FPID controller in both servo and regulatory mode. SOFC has offered 23.43 %, 60.50 % and 36.20 % improvements in IAE link-1, IAE link-2 and cost function for trajectory tracking and 35.21 %, 51.34 % and 39.63 % improvements in IAE link-1, IAE link-2 and cost function for disturbance rejection respectively.

Jitendra Kumar, Vineet Kumar, K. P. S. Rana

### Comparative Analysis of Different Nature Inspired Optimization Algorithms for Estimation of 3D Chaotic Systems

Among various nonlinear systems, parameter identification of chaotic systems turns out to be a very challenging task because of their complex and unpredictable nature. The control and synchronization of chaotic systems remains incomplete until the parameters of the chaotic systems are known. Traditionally, the trend has been to estimate the parameters using gradient based search methods which suffer from premature convergence and trapping in local minima. This chapter presents an optimization based scheme for estimation of the parameters of two chaotic systems namely Lorenz and Rossler, using two recently developed bio-inspired optimization algorithms, i.e. cuckoo search algorithm (CSA) and flower pollination algorithm (FPA). CSA is based on mimicking the breeding behavior and hostile reproduction strategies of cuckoo with the effective use of levy flight for providing global optimization while FPA is based on the natural process of flowering plants due to self and cross pollination using both levy flight strategies for global convergence and random walk for local convergence. The performance of these optimization algorithms, for efficient estimation of the parameters of chaotic system, is compared in terms of the resulting integral of absolute error (IAE). Simulation results demonstrated the effectiveness of CSA in offline 3D parameter estimation of the considered two chaotic systems over the FPA. The minimum fitness offered by FPA is 2.4E-03 and 5.03E-06 and by CSA it is 7.92E-06 and 1.31E-07 for the parameter estimation of the 3D Lorenz and Rossler chaotic system, respectively.

Sreejith S. Nair, K. P. S. Rana, Vineet Kumar

### Swarm Intelligence PID Controller Tuning for AVR System

The voltage regulator is designed to automatically maintain a constant voltage level in the power system. It may be used to regulate one or more AC or DC voltages in power systems. Voltage regulator may be designed as a simple “feed-forward” or may include “negative feedback” control loops. Depending on the design, it may use an electromechanical mechanism, or electronic components. The role of an AVR is to keep constant the output voltage of the generator in a specified range. The PID controller can used to provide the control requirements.The chapter discusses the methods to get the best possible tuning controller parameters for an automatic voltage regulator (AVR) system of a synchronous generator. It was necessary to use PID controller to increase the stability margin and to improve performance of the system. Some modern techniques were defined. These techniques as Particle Swarm Optimization (PSO), also it illustrates the use of a Adaptive Weight Particle Swarm Optimization (AWPSO), Adaptive Acceleration Coefficients based PSO, (AACPSO), Adaptive Acceleration Coefficients based PSO (AACPSO). Furthermore, it introduces a new modification for AACPSO technique, Modified Adaptive Acceleration Coefficients based PSO (MAAPSO) is the new technique which will be discussed inside the chapter, A comparison between the results of all methods used will be given in this chapter. Simulation for comparison between the proposed methods will be displayed. The obtained results are promising.

Naglaa K. Bahgaat, M. A. Moustafa Hassan

### Discrete Event Behavior-Based Distributed Architecture Design for Autonomous Intelligent Control of Mobile Robots with Embedded Petri Nets

This chapter presents a design methodology of discrete event distributed control architecture for autonomous mobile robot systems. A modular, behavior-based distributed software architecture is presented on a hierarchical distributed microcontroller based hardware structure for intelligent control of mobile robots. Some intelligent behaviors, such as wall following, obstacle rounding, target seeking, and local environment mapping, have been implemented using sensor control modules such as multiple infrared range finding sensor modules and motion control modules to detect walls and obstacles in the surroundings of a mobile robot, based on environment features such as lines and corners estimated using a set of range sensors and a vision sensor. Upon these behavior modules, a Petri net based approach was applied to coordination of several concurrent activities of modules for the high-level tasks such as sensory navigation in unknown environments. Task specification implies the definition of a control program composed of behavior commands, which are not expressed in a sequential fashion but implicating parallel processing control. The net model can be directly obtained from the system requirements specification of each particular application. Thus, the remaining levels of the control structure are common to a wide range of applications. The Petri net based approach validates the implementation of synchronization and coordination in discrete event behavior-based control. Behavior modules are composed to design more complex modules according to applications. The detailed function of each control module is specialized according to the application, so that new control strategies can be easily embedded in the control modules for real-time performance of robotic actions. Compared to hand–written coding in robot program, because of explicit representation of robotic actions, behaviors and tasks, the system design procedure facilitates the understanding of the interaction among the different processes that might be present in the mobile robot control system. Consequently, it is easy and computationally inexpensive to design, write, and debug planned tasks. Besides it is possible to verify structural and behavioral properties of these programs owing to formal specification.

Gen’ichi Yasuda

### Indoor Thermal Comfort Control Based on Fuzzy Logic

Control and monitoring of indoor thermal conditions represent crucial tasks for people’s satisfaction in working and living spaces. In the first part of the chapter we address thermal comfort issues in a working office scenario. Among all standards released, predicted mean vote (PMV) is the international index adopted to define users thermal comfort conditions in moderate environments. In order to optimize PMV index we designed a novel fuzzy controller suitable for commercial Heating, Ventilating and Air Conditioning (HVAC) systems. However in a residential scenario it would be extremely expensive to gather real time measures for PMV computation. Indeed in the second part of the chapter we introduce a novel approach for residential multi room comfort control based on humidex index. A fuzzy logic controller is introduced to reach and maintain comfort conditions in a living environment. Both control systems have been experimentally tested in the central east coast of Italy. Temperature regulation performances of both approaches have been compared with those of a classical PID based thermostat.

Lucio Ciabattoni, Gionata Cimini, Francesco Ferracuti, Gianluca Ippoliti, Sauro Longhi