Hyperchaotification and Synchronization of Chaotic Systems

This paper addresses the hyperchaotification and the synchronization of chaotic systems. A nonlinear state feedback controller was designed to generate hyperchaos from the original chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents, bifurcation diagram and phase diagrams. Furthermore, effective active controllers are designed for synchronizing the obtained hyperchaotic system with different chaotic systems. To illustrate the effectiveness of the proposed approach, numerical simulation results obtained with the Lü and Qi chaotic systems are given.


Introduction
Chaos has been extensively studied within the scientific, engineering and mathematical co mmunit ies as an interesting complex dynamic phenomenon. Recently, the tradit ional trend of understanding and analysing chaos has evolved to a new phase of investigation: controlling and creating chaos. More specifically, when chaos is useful, it is generated intentionally. However, when chaos is harmful, it is controlled [1][2][3][4].
Indeed, several studies have showed that chaos can be useful or has great potential in many disciplines and most of the developed methods concern with the chaotic synchronization [5][6][7][8]. Pecora and Carro ll [5] suggested that the phenomenon of chaos synchronism may serve as the basis for new ways for achieving secure commun ication. Since, many techniques have been proposed in order to h ide the contents of a message by explo iting chaotic systems. Perez and Cerderia have proved that messages masked by simp le chaotic processes can be easily extracted once intercepted [9].
After, Pecora proved that this problem can be solved by using higher dimensional hyperchaotic systems [10]. This consideration has led to the development o f interesting techniques for hyperchaos synchronization [11,12].
Hyperchaotic systems have the characteristics of high security and high efficiency. They can be broadly applied in nonlinear circu its, secure communications, bio logical systems and many other fields.
In this paper, we are interested in the hyperchaotification and synchronization of chaotic systems. The hyperchaotification is obtained using a delay feedback control algorith m. It is based on the idea that hyperchaotic systems are usually defined as chaotic systems with more than one positive Lyapunov exponent. A closed form expression was provided for the controller, in terms of the system state vector and a set of Lyapunov exponents. Simultaneously, the controlled system is synchronized with a second chaotic system, by applying an algorith m based on the active control theory. The paper is organized as follows: In section 2, we introduce the hyperchaotification algorith m and its application to the Lü system. In section 3, the proposed synchronization algorith m is detailed on chaotic and hyperchaotic systems and we provide some simulat ion results obtained with the Lü and Qi systems. These results illustrate the effectiveness of the proposed procedure.

Hyperchaotification of Lü System
The Lü system is described by the following equations: Where a , b and c represent the system parameters The system o f d ifferential equations is integrated using the fourth order Runge-Kutta method.
For: 36 = a We deduce that for these values, the system is chaotic since one of the exponents is positive and the system exhib its a chaotic behaviour, as shown in Fig. 1.
For the hyperchaotification of a chaotic system, the two following conditions must be verified : The dimension of the system must be at least equal to 4 and the order of the state equation must be at least 2.
The system must have at least two positive Lyapunov exponents and the sum of all the exponents must be negative.
For this, the hyperchaotification of the Lü chaotic system consists to increase the dimension of the system by adding an equation representing a state feedback controller in the state equations system. So me authors suggested to construct a hyperchaotic attractor of the Lü system by adding the state feedback controller u on the state x . In our case, we choose to introduce it on y, as follows: ( ) x a y x y xz cy u z xy bz u yz du The fixed points are obtained by solving: And the jacobian matrix of the system is given by: This implies that the stability around the fixed points is function of (a, b, c) and also d.
We can conclude that for some values of d, this method of control stabilizes the in itial chaotic system. Ho wever, if this parameter continues to increase, a hyperchaotic behaviour appears. In Fig. 5 we can notice that the obtained hyperchaotic attractor has a similar fo rm that the original Lü system with the appearance of additional branches which characterizes the hyperchaotic dynamic

Synchronization of Hyperchaotic and Chaotic Systems
Many effective methods have been presented to synchronize chaotic systems. Synchronization is always done between a system designed as master and another as slave. The principle of synchronization is to apply on the slave a control function, such as the error between the two systems tends to zero. The problem can then be expressed as a problem of control that consists of minimizing the error between the master and the slave by applying the control law. In our case we use an active control algorith m.
In the following, we consider the synchronization of the hyperchaotic Lü system with a chaotic system, by applying an algorithm based on the active control theory. The hyperchaotic system is considered as master and the chaotic system as slave.

Synchronization of the Hyperchaotic LÜ System and the 4D Chaotic LÜ S ystem
The master system is given by: The control function is co mposed of two parts: one non linear for eliminating non linear term and a linear part to ensure the stability of the obtained system.
The error between the master and the slave is given by: ( 1) ( 1) In this case, all the eigenvalues are equals to -1 and the error 0 e → when t → +∞ . For the simu lation, we choose the master system as the new hyperchaotic Lü system with d 1 =1.8. The same system with d 2 =-5 is considered as the chaotic slave system For the others parameters, we assume that: a 1 =a 2 =a=36, b 1 =b 2 =b=3, c 1 =c 2 =c=20. Fig.(6-a) to Fig.(6-d) represent the results of the synchronizat ion when the control is actived at t=2s and initial con- The master system is the same as for the previous case and the slave system is given by: The error between the master and the slave is given by: The control law consists of two parts: A part to eliminate the nonlinear terms and another to stabilize the resulting linear system 1 1 So, the controlled system is as follo ws: In Fig. 7(a-c), the trajectories of the two systems are in itially co mpletely different due to the sensitivity to the init ial conditions. Once activated the control at t=2s, the two systems take a short time to be perfectively synchronized. The trajectory of the chaotic slave system become the same as that of the master hyperchaotic system

Synchronization of the Hyperchaotic Lü System and the Qi Chaotic System
In this examp le, the master system is the hyperchaotic Lü system, and the slave system is the Qi chaotic system. This implies that when the two systems are synchronized, the Qi chaotic system will fo llow the same trajectories as those of the hyperchaotic Lü system.
As for the previous examp le, we define the master system: The active control function applied on the slave is: ( 1)

Conclusions
In this paper, we introduced the generation of new hyperchaotic Lü system. Dynamical behaviours of the system are exp lored by calcu lating the Lyapunov exponents and the phase diagram. The synchronization of the obtained hyperchaotic system with a chaotic system is possible using an active control algorithm. In addition to its efficiency, this method is easy to implement and achieves the synchronization of t wo systems comp letely d ifferent, in a reduced time. The stability is guaranteed since the control law ensures that the eigenvalues of the system are always in the left part of the complex p lan.