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In this paper, an adaptive control scheme is developed to study the hybrid synchronization behavior between two identical and different hyperchaotic systems with unknown parameters. This adaptive hybrid synchronization controller is designed based on Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. The adaptive hybrid synchronization between two identical systems (hyperchaotic Chen system) and different systems (hyperchaotic Lorenz and hyperchaotic

Chaos is an omnipresent phenomenon. Scientists who understand its existence have been struggling to control chaos to our benefit. There is a great need to control the chaotic systems as chaos theory plays an important role in industrial applications particularly in chemical reactions, biological systems, information processing and secure communications [1-3]. Many scientists who are interested in this field have struggled to achieve the synchronization or anti-synchronization of different hyperchaotic systems. Therefore due to its complexity and applications, a wide variety of approaches have been proposed for the synchronization or anti-synchronization of hyperchaotic systems. The types of synchronization used so far include generalized active control [4-8], nonlinear control [9,10], and adaptive control [11-19].

The co-existence of synchronization and anti-synchronization, known as hybrid synchronization, has good application prospects in digital communications. Therefore it attracted a lot of attention in recent years. In hybrid synchronization scheme, one part of the system is anti-synchronized and the others are completely synchronized so that complete synchronization and anti-synchronization co-exist in the system. The co-existence of CS and AS may enhance security in communication and chaotic encryption schemes. Li [

Zhang and Lű [

In the first part of this section, we set up the problem and present novel adaptive hybrid synchronization scheme with parameter update law. By using Lyapunov stability theory we show the co-existence of hybrid synchronization between two systems described below. In the second part of this section we briefly describe the two systems used for further analysis.

Consider the master chaotic system in the form of

where is the state vector, is the unknown constant parameter vector of the system, is an matrix, is an matrix whose elements. The slave system is assumed by:

where is the state vector, is the unknown constant parameter vector of the system is an matrix, is an matrix whose elements, and is control input vector. If we divide the master and the slave systems into two parts, then system (1) can be written as:

and the slave system (2) can be written as

Let and be the synchronization and the anti–synchronization error vector’s respectively. Our goal is to design a controller such that the trajectory of the response system (5)-(6) with initial conditions can asymptotically approach the drive system, (3)-(4), with initial condition And finally implement the hybrid synchronization such that, and the anti-synchronization such that

where is the Euclidean norm.

Theorem: If the nonlinear control is selected as:

and adaptive laws of parameters are taken as:

then the response system (5)-(6) can synchronize and anti-synchronize the drive system (3)-(4) globally and asymptotically, where and are respecttively, estimations of the unknown parameters and

Proof: From Equations (3)-(6), we get the error dynamical systems as follows:

where

Let and .

If a Lyapunov function candidate is chosen as

The time derivative of V along the error dynamical system is given by:

Since is positive definite, and is negative semi-definite, it follows that from the fact that

It can easily be seen that From Equation (9) have. Thus, by Barbalat’s lemma, we have Thus the response system (2) can be synchronized and anti-synchronized the drive system (1) globally and asymptotically. This completes the proof.

The hyperchaotic Chen system [26,27] is given by:

where and are state variables, and , and are real constants. When system (13) is chaotic, when , system (13) is hyperchaotic.

The hyperchaotic Lorenz system [28,29] is described by

where, and are state variables, and are real constants. When and system (14) has hyperchaotic attractor.

The hyperchaotic Lű system [

where and are state variables, and are real constants. When system (15) has hyperchaotic attractor.

In order to observe the efficacy of our proposed method, we used two hyperchaotic Chen systems where the master system is denoted with the subscript 1 and the response system having identical equations denoted by the subscript 2. The two systems are defined below.

and

where are four control functions to be designed. For the hybrid synchronization, we define the state errors between the response system that is to be controlled and the controlling drive system as . The error system is given by

Now our goal is to find proper control functions and parameter update rule, such that system (17) globally hybrid synchronizes system (16) asymptotically. i.e., where

If the two systems are without controls and the initial condition is:

then the trajectories of the two systems will quickly separate each other and become irrelevant. However, when appropriate controls are applied the two systems will approach hybrid synchronization for any initial conditions. We shall propose the following adaptive control law for system (17).

where are the estimates of respectively. Now, let us choose a controller and parameters update law as follows:

and the parameter update rule.

Consider the following Lyapunov function

Consider the following Lyapunov function

Then the time derivative of along the trajectories of Equation (18) is:

Since is positive definite function and is negative definite function, it translates to based on the Lyapunov stability theorem [

To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation results for hyperchaotic Chen system. In the numerical simulations, the fourth-order Runge-Kutta method is used to solve the systems with time step size 0.001. For these numerical simulations, we used the initial conditions, and . Hence, the error system has the initial values and The unknown parameters are chosen as and such that the hyperchaotic Chen system exhibits chaotic behavior. Hybrid synchronization of systems (16) and (17) via adaptive control laws (Equations (19) and (20)) with the initial estimated parameters and are shown in Figures 1 and 2. Figures 1(a) and (d) display state trajectories of drive system (16) and the response system (17).

In order to observe the hybrid synchronization behavior between hyperchaotic Lorenz system (15) and hyperchaotic Lű system (14), we assume that hyperchaotic Lorenz system with four unknown parameters is the drive system and hyperchaotic Lű system with four unknown parameters is the response system. The drive and response systems are defined as follows:

and

where are four control functions to be designed. For the hybrid synchronization, we define the state errors between the response system that is to be controlled and the controlling drive system as

The error system is given by

Now our goal is to find proper control functions and parameter update rule, such that system (17) globally hybrid synchronizes system (16) asymptotically. i.e. where

. If the two systems are without controls and the initial condition is

then the trajectories of the two systems will quickly separate each other and become irrelevant. However, when appropriate controls are applied the two systems will approach hybrid synchronization for any initial conditions. We shall propose the following adaptive control law for system (23). We define the parameters error and

where

and are the estimates of and respectively. Now, let us choose a controller and parameters update law as follows:

and the parameter update rule

Consider the following Lyapunov function

Then the time derivative of along the trajectories of Equation (24) is

Since is positive definite function and is negative definite function, it translates to based on the Lyapunov stability theorem [

To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation result for the hybrid synchronization between hyperchaotic Lorenz

system and hyperchaotic Lű system. In the numerical simulations, the fourth-order Runge-Kutta method is used to solve the systems with time step size 0.001. For this numerical simulation, we assume that the initial condition, and is employed. Hence the error system has the initial values and. The unknown parameters are chosen as

and

in simulations so that both the systems exhibits a hyperchaotic behavior. Hybrid synchronization of and (26) with the initial estimated parameters

and are shown in Figures 3 and 4.

as.

In this paper, we discussed the problem of adaptive hybrid synchronization of hyperchaotic systems with fully unknown parameters. On the basis of the Lyapunov stability theory and the adaptive control theory, a new adaptive hybrid synchronization control law and a novel parameter estimation update law are proposed to achieve hybrid synchronization between the two identical and different hyperchaotic systems with uncertain parameters. This shows that our proposed method has strong robustness. Finally, the simulation results are presented to show the effectiveness of this approach.