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In this paper, we consider the chaos control for 4D hyperchaotic system by two cases, known & unknown parameters based on Lyapunov stability theory via nonlinear control. We find that there are two cofactors that have an effect on determining any case to achieve the control, the two cofactors are proposed in the control and the matrix that produce from the time derivative of Lyapunov function. In adding, we find some weakness cases in Lyapunov stability theory. For this reason, we design with only one controller and perform a simple change in this control in order to recognize the difference between these cases although all of the controllers are almost similar.

Chaos phenomenon was firstly observed by Lorenz in 1963 [

Many different techniques for chaos control and synchronization have been developed, such as a linear feedback method, active control approach, adaptive technique, time delay feedback approach, and back stepping method. Among them, nonlinear control is an effective method to control chaos [

In dynamical systems, there are three types of parameters, known, unknown and uncertain parameters. However, some of the previous works achieved control and synchronization with unknown parameters only and another some achieved control and synchronization with known parameters only. In this paper, we perform chaos control for two cases and we find that the proposed control plays an important and active role to determine the case as well as the matrix of time derivative for Lyapunov function, where the strategy of this paper is based on designing only one control. We also perform some simple change into this control, study probability of suppression for each control by using Lyapunov stability theory and get three cases. In first case, we can achieve control directly when parameters are unknown; in second type, we need to modify in order to achieve control with known parameters; finally in third type, it is imposable to perform the control.

Although the accuracy of Lyapunov stability theory is not neglected and the nonlinear parts have been successful in the treatment of the first two types, but it failed to treat the third type. So, we refuge to use the linear approximation method to treat the problem and the weaknesses.

Briefly, this study poses three fundamental questions. First, when can we achieve chaos control with known parameters? Second, when can we achieve chaos control with unknown parameters? And third, how can we distinguish between these two cases? This paper begins with the suggestion of a new method that will answer these questions.

In this section, we describe the problem formulation for the chaos control and chaos synchronization for hyperchaotic systems and our methodology using nonlinear control by basing on the Lyapunov stability theory.

Let us consider the hyperchaotic system in the following form:

where

If we add the controller

The aim of the control problem is to design a feedback controller U such that

But, in synchronization problem we needed two system, drive system and response system. Let us consider the system (1) as the drive system and the response system is given by the following forms:

where

The error dynamics for the synchronization can be expressed as

where

It is clear that the problem of synchronization between the drive and response systems is replaced by the equivalent problem of stabilizing the system (4) using a suitable choice of the control U [

In unknown parameter, we assume that the Lyapunov function is always formed as

According to the above discussion, and based on Lyapunov stability theory, we can obtain the following conclusion.

Corollary 1. To determine the chaos control with known and unknown parameters we based on two cofators:

a) Controller design, b) the matrix Q.

If we get the matrix Q as:

a) Identity or diagonal matrix. Then, we choose chaos control with unknown parameters; b) not diagonal matrix. Then, we choose chaos control with known parameters (in order to translate not diagonal matrix to diagonal we must give the value of parameters and make modify on the matrix P).

In this section, we achieve controlling problems with known and unknown parameters for the hyperchaotic system [

where

In order to control above hyperchaotic system to zero, the feedback controllers of

in which

In the following theorem, we proposed nonlinear control with unknown parameters to control system (6).

Theorem 1. The controlled hyperchaotic system (6) will achieve globally asymptotically stable with the following nonlinear controllers:

Proof. Substituting the controllers (7) in the system (6), we have

According to the formulation (1), the above system can be rewritten as:

To achieve the control of this system, there are two methods, Lyapunov stability theory and linearization method.

Based on the Lyapunov stability theory and corollary 1, we construct the following Lyapunov candidate function

where

Now, by controllers (7), the time derivative of the Lyapunov function is:

Here

Also, by using the linearization method, we have the characteristic equation as:

To solve this equation with unknown parameters, we needed to analytically (Theoretical) methods such that Gardan method and Routh-Hurwitz method, while using the numerical methods when we knew these parameters.

By Routh-Hurwitz method, Equation (12) has all roots with negative real parts if and only if

If we make simple change into control (7) i.e. change only second equation to become the following forms:

and used the same of the Lyapunov function in Equation (10). Then, we get the time derivative for the Lyapunov function as the following:

where

By this control, we obvious the matrix

Theorem 2. The controlled hyperchaotic system (6) with nonlinear control (13) is globally asymptotically stable.

Proof. The system (6) with control (13) becomes:

This system can be reformulated in the following form:

Now, according to the Lyapunov second method, Let us modify the Lyapunov function by the following form i.e. modify the matrix p to get the matrix

where

So, we have the time derivative of the Lyapunov function as:

where

Consequently, we translate the matrix

On the other hand, the control problem for a system (6) with control (13) can be achieved by linearization method. Then, we have the characteristic equation forms:

Since the parameters are known. So, we have

and the roots of the above equation are

Remark1. We founded the roots of Equation (20) by numerical methods. Also, we can use Gardan and Routh-Hurwitz methods for it.

Obviously, from theorem 1, the matrix

Theorem 3. If the nonlinear controllers are proposed as:

i.e. simple change in four equation for control (7). Then, the zero solution of the controlled hyperchaotic system (6) can’t convergent by Lyapunov stability theory and is globally asymptotically stable by linearization method.

Proof. Substituting the controllers (21) in the system (6), we have

Also system (22) can be rewritten (According to the formulation (1)) as:

To check the control of this system by using Lyapunov stability theory, we can construct a Lyapunov function as

Here

So it is impossible to turn this matrix

Since the parameters are known, we have

And the roots of this equation are

In this section, we consider the synchronization problem of the hyperchaotic system (5) with known and unknown parameters using corollary 1. and how we can apply this corollary to determine between them,

Let us consider the hyperchaotic system (5) as the drive system, and the controlled hyperchaotic system (6) as the response system.

Subtracting system (5) from the system (6), we obtain the error dynamical system between the drive system and the response system which is given by:

where

System (27) describes the error dynamics according to formulation 4.

Theorem 4. The zero solution of the error dynamical system (27) is asymptotically stable if nonlinear control is designed as following:

Proof. Substituting the controllers (28) in the system (27), we have

According to the formulation (1), the above system can be rewritten as:

Based on the Lyapunov stability theory, we construct the following Lyapunov candidate function

And the time derivative of the Lyapunov function is:

Here

As well, by using the linearization method, we have the characteristic equation as:

By Routh-Hurwitz method, Equation (33) has all roots with negative real parts if and only if

Based on the previously discussed in Section 3 to make simple change into a new control (change only in first equation for control 28) to become as:

and used the same of the Lyapunov function in Equation (31). Then, we take the time derivative for the Lyapunov function as the following:

where

Obviously, the matrix

Theorem 5. The error dynamical system (27) with control (34) is globally asymptotically stable.

Proof. The system (27) with control (34) becomes as:

This system can be reformulated in the following form:

Now, according to the Lyapunov second method, Let us modify the Lyapunov function of the following form:

where

So, we have the time derivative of the Lyapunov function as:

where

We translate the matrix

which gives asymptotic stability of the system (27) by Lyapunov stability theory. This means that the controller proposes is achieved the suppressed of system (27).

On the other hand, the control problem for a system (27) with control (34) can be achieved by linearization method. Then, we have the characteristic equation forms:

Since the parameters are known, we have

And the roots of this equation are

In adding, if we choose nonlinear control (simple change into four equations for control (34)) as:

with the matrix p then we have the matrix

To translate the matrix

where

Obviously, from theorem 4, we can perform the controlling of error dynamics system (27) by using controller (28) with matrix p directly, while in theorem 5 we must modify the matrix P to become

Theorem 6. If the nonlinear controllers are proposed as:

We multiplied the parameter d by number 2 in equation four for control (28). Then, it is impossible to perform the synchronization by Lyapunov stability theory.

Proof. The system (27) with control (44) becomes as

This system can be reformulated in the following form:

Now, to check the control of this system by using Lyapunov stability theory, we can construct a Lyapunov function as

Here,

So it is impossible to transient the matrix

Since the parameters are known, we have

and the roots of this equation are

Verification, we can be used the numerical simulation to validate these proposed controls, we choose the parameters

system (5) and the response system (6) with control (28), (34) and control (44) respectively.

In this paper, we study the chaos control for 4D hyperchaotic system based on Lyapunov stability theory, where this method is effective and accurate in finding stability of systems, and in view of dealing with nonlinear parts of systems and not neglecting those parts which support the strength and accuracy.

Nevertheless, it loses this property in some time. As the case of the system in this paper, we design the control to ensure the survival of nonlinear parts in the system. And in some cases, it can suppress without knowing the parameters of the system and the other cases. We must know that the parameters and third case can’t be suppressed. Then, the Lyapunov stability theory will be failed in sometimes. Therefore, we use the linear approximation method to ensure the validity of this proposed control. We have succeeded in achieving control, and we find that the simple difference in the control is responsible to get these three cases. Through this method, we can treat every case when the nonlinear parts have no effect on the system. Finally, numerical simulations show the effectiveness of the proposed chaos control and synchronization schemes.

MaysoonM. Aziz,Saad FawziAl-Azzawi, (2016) Control and Synchronization with Known and Unknown Parameters. Applied Mathematics,07,292-303. doi: 10.4236/am.2016.73026