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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 515106, 18 pages

http://dx.doi.org/10.1155/2013/515106

## Chaotic Control and Generalized Synchronization for a Hyperchaotic Lorenz-Stenflo System

^{1}Department of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China^{2}School of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China

Received 20 July 2013; Accepted 23 September 2013

Academic Editor: Yong Ren

Copyright © 2013 Yin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is devoted to investigate the tracking control and generalized synchronization of the hyperchaotic Lorenz-Stenflo system using the tracking model and the feedback control scheme. We suppress the chaos to unstable equilibrium via three feedback methods, and we achieve three globally generalized synchronization controls. Novel tracking controllers with corresponding parameter update laws are designed such that the Lorenz-Stenflo systems can be synchronized asymptotically. Moreover, numerical simulations are presented to demonstrate the effectiveness, through the contrast between the orbits before being stabilized and the ones after being stabilized.

#### 1. Introduction

Study of chaotic control and generalized synchronization has received great attention in the past several decades [1–10]; many hyperchaotic systems have been proposed and studied in the last decade, for example, a new hyperchaotic Rössler system [4], the hyperchaotic L system [5], Chua’s circuit [6], the hyperchaotic Chen system [7, 8], and so forth. Hyperchaotic system has been proposed for secure communication and the presence of more than one positive Lyapunov exponent clearly improves the security of the communication scheme [9–12]. Therefore, hyperchaotic system generates more complex dynamics than the low-dimensional chaotic system, which has much wider application than the low-dimensional chaotic system.

Until now, a variety of approaches have been proposed for the synchronization of low-dimensional chaotic systems, including Q-S method [13, 14], active control [15, 16], adaptive control [17–24], and time-delay feedback control [25]. Recently, Stenflo [26] presented a new hyperchaotic Lorenz-Stenflo (LS) system: In (1), , , , and are the state variables of the system and , , , and are real constant parameters. System (1) is generated from the originally three-dimensional Lorenz chaotic system by introducing a new control parameter and a state variable .

In this paper, we will consider chaos control and generalized synchronization related to hyperchaotic Lorenz-Stenflo system. we found that the feedback control achieved in the low-dimensional system like many other studies of dynamics in low-dimensional systems. We suppress the hyperchaotic Lorenz-Stenflo system to unstabilize equilibrium via three control methods: linear feedback control, speed feedback control, and doubly-periodic function feedback control. By designing a nonlinear controller, we achieve the generalized synchronization of two Lorenz-Stenflo systems up to a scaling factor. Moreover, numerical simulations are applied to verify the effectiveness of the obtained controllers.

#### 2. The Hyperchaotic Lorenz-Stenflo System

In the following we would like to consider the hyperchaotic cases of system (1). When , , , and , system (1) exhibits hyperchaotic behavior. Simulated results are depicted in Figures 1 and 2. Figures 1(a)–1(d) depict the projection of the chaotic attractor in different spaces; Figures 2(a)–2(d) depict the states of system (1) before being stabilized, respectively. The volume of the elements in the phase space and the divergence of flow (1) are defined by where .

System (1) is dissipative when . Moreover, an exponential contraction rate is given by It is clear that , which implies that the solutions of system (1) are bounded as . It is easy to find the three equilibria ,

To determine the stability of the equilibria point , evaluating the Jacobian matrix of system (1) at yields If , , , and , the four eigenvalues of the characteristic polynomial of Jacobian matrix (5) are Thus, the equilibria is a saddle point of the hyperchaotic system (1). The Jacobian matrix of system (1) at yields If , , , and , the four eigenvalues of the characteristic polynomial of Jacobian matrix (7) are Thus, the equilibria are unstable, and is similar.

#### 3. The Hyperchaotic Control for Lorenz-Stenflo System

In this section, we control the hyperchaotic system (1) such that all trajections converge to the equilibrium point . The controlled hyperchaotic Lorenz-Stenflo system is given by where , , , and are external control inputs which will be suitably derived from the trajectory of the chaotic system (1), specified by to the equilibrium of uncontrolled system .

##### 3.1. Linear Function Feedback Control

For the modified hyperchaotic Lorenz-Stenflo system (9), if one of the following feedback controllers is chosen for the system (9), then , , , , and where ’s are feedback coefficients. Therefore controlled system (9) is rewritten as whose Jacobian matrix is The characteristic equation of is where , , , , and .

The abbreviated characteristic equation is where , , and , .

According to the Routh-Hurwitz criterion, constraints are imposed as follows:

This characteristic polynomial has four roots, all with negative real roots, under the condition of , , , and . Therefore, the equilibria are the stable manifold and the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable. The concrete dynamics of (9) can be demonstrated by Proposition 1, while the Maple program is demonstrated in Appendix A.

Proposition 1. *If one chooses the control coefficients , , , and and the parameters , , , and , the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium .*

*Proof. *When the parameters were selected by the above value, we obtain the Jacobian matrix
The characteristic equation of is given by
According to Appendix A, we easily obtain
which yields the eigenvalues via the compute simulation
Thus the zero solution of system (9) is exponentially stable, Proposition 1 is proved.

Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (1) using the fourth-order Runge-Kutta scheme with time step 0.01. The parameters and the corresponding feedback coefficients are given by the above value. The initial values are taken as . The behaviors of the states of the controlled chaotic Lorenz-Stenflo system (1) with time are displayed in Figures 3(a)–3(d).

##### 3.2. Speed Function Feedback Control

Suppose that ; and are of the speed forms and , where and are speed feedback coefficients. Therefore the controlled chaotic system (9) is rewritten as The Jacobian matrix is The characteristic equation of is

Proposition 2. *If one chooses the control coefficients: , and the parameters , , , and , the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium .*

Similar to Proposition 1, the proof of Proposition 2 is straightforward and thus is omitted.

In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix The characteristic equation of changes the following: The eigenvalues of the above equation are Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (20) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as . The behaviors of the states of the controlled chaotic Lorenz-Stenflo system (20) with time are displayed in Figures 4(a)–4(d).

##### 3.3. The Doubly Periodic Function Feedback Control

Suppose that , , and ; is of the doubly periodic function , where is speed feedback coefficients and is the modulus of Jacobi elliptic function. Therefore the controlled chaotic system (9) is rewritten as The Jacobian matrix is The characteristic equation of is

Proposition 3. *If one chooses the control coefficients , and the parameters , , , and , the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium .*

The proof of Proposition 3 is the same as Proposition 1, which is straightforward and thus is omitted.

In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix The characteristic equation of changes the following: The eigenvalues of the above equation are Thus the zero solution of system (9) is exponentially stable; Proposition 3 is proved.

Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (25) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as . The behaviors of the states of the controlled chaotic Lorenz-Stenflo system (25) with time are displayed in Figures 5(a)–5(d).

#### 4. Globally Exponential Hyperchaotic Projective Synchronization Control

Consider two chaotic systems given by where , , , , and . comprises the set of non negative real numbers. Assume that (31) is the master system, (32) is the slave system, and is the control vector. Let the error state be Then the error dynamics of are defined by The slave and master systems are said to be exponential, hyperchaotic, projective and synchronized if, for all, , and , as .

Lemma 4 (see [27, 28]). *The zero solution of the error dynamical system (34) is globally and exponentially stable; the master-slave systems (31) and (32) are globally and exponentially projective, synchronized, if there exists a positive definite quadratic polynomial such that . Moreover, the following negative Lyapunov exponent estimation for the error dynamical system (34) holds:
**
where and are both positive definite matrices, and stand for the minimum and maximum eigenvalues of the matrix , respectively, and denotes the minimum eigenvalue of the matrix .*

In the following, we consider the hyperchaotic system (1) as a master system: and the system related to (36), given by as a slave system, where the subscripts “” and “” stand for the master system and slave system, respectively. Let the error state be Then the derivative of along the trajectories of (36) and (37), we obtain the error system: To demonstrate the synchronization control between systems (36) and (37), we have the following cases based on [29–44].

*Case 1* (modified projective synchronization control).

Theorem I. *When , and . For the hyperchaotic system (1), if one of the following families of feedback controllers is given for the slave system (36):
**
there exist many possible choices for , , , and . Then the zero solution of the error dynamical system (39) is globally and exponentially stable, and thus globally exponential modified projective synchronization can be achieved.*

The concrete proof of Theorem I can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (42) with Runge-Kutta integration method. The initial values of the drive system and response system are , , , and and , , , and respectively. The parameters are chosen to be , , , , and so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The numerical simulation of the master and the slave systems without active synchronization control law is shown in Figures 6(a)–6(d). The diagram of the solutions of the master and the slave systems with feedback control law is presented in Figures 7(a)–7(d). The synchronization errors are shown in Figures 8(a)–8(d). Figures 9(a)–9(d) depicts the projection of the synchronized attractors. Figures 10(a)–10(f) depicts the phase portraits of the synchronized attractors.

*Case 2* (generalized projective synchronization control).

Theorem II. *When , ’s are different, and . For the hyperchaotic system (1), if one of the following families of feedback controllers is given for the slave system (36):
**
there exist many possible choices for , , , and . Then the zero solution of the error dynamical system (39) is globally and exponentially stable, and thus globally exponential generalized projective synchronization.*

The concrete proof of Theorem II can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (39) with Runge-Kutta integration method. The parameters are chosen to be , , , and . The initial values and others parameters are the same as the above cases so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The diagram of the solutions of the master and the slave systems with feedback control law is presented in Figures 11(a)–11(d). The synchronization errors are shown in Figures 12(a)–12(d).

*Case 3* (function synchronization control [45–58]).

Theorem III. *When , , , , , , . For the hyperchaotic system (1), if one of the following families of feedback controllers is given for the slave system (36):
**
where , , , and , then the zero solution of the error dynamical system (39) is globally stable, and thus global function, projective synchronization occurs between the master systems (36) and (37).*

The concrete Proof of Theorem III can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (39) with Runge-Kutta integration method. The initial values and the parameters are the same as the above cases so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The diagram of the solutions of the master and the slave systems with active control law is presented in Figures 13(a)–13(d). The synchronization errors are shown in Figures 14(a)–14(d).

#### 5. Summary and Conclusions

In this paper, we have introduced the tracking control and generalized synchronization of the hyperchaotic system which is different from the Lorenz-Stenflo attractor. We suppress the chaos to unstabilize equilibrium via three feedback methods, and we achieve three globally generalized synchronization controls of two Lorenz-Stenflo systems. As a result, some powerful controllers are obtained. Then, we investigate the hyperchaotic system applying the complex system calculus technique. Moreover, numerical simulations are used to verify the effectiveness of our results, through the contrast between the orbits before being stabilized and the ones after being stabilized.

#### Appendices

#### A. The Maple Program

*restart: with(student): with(PDEtools): with(linalg): with (LinearAlgebra)*: ,; ; ; ; ; ; ; ,; ; ; ; ; ; ; ;
.

#### B. The Proof of Theorem

*Proof of Theorem I. *Consider the controller (40) and choose the following :
Then the system (39) is reduced into
Let us consider the Lyapunov function for the system (B.2) as follows:
In addition, the derivative of has the form
which is negatively defined. So (B.2) is asymptotically stable. This implies that the two Lorenz-Stenflo hyperchaotic systems are projective and synchronized.

*Proof of Theorem II. *Consider the controller (41) and choose as follows:
Then the following steps are the same (B.2), (B.3), and (B.4). So we know that the two Lorenz-Stenflo hyperchaotic systems are modified, projective, and synchronized.

*Proof of Theorem III. *Consider the controller (42) and choose the following positive definite, quadratic form of Lyapunov function:
We differentiate and substitute the trajectory of system (39) which yields
where , which implies that the conclusion of Theorem III is true. So (39) is asymptotically stable. This implies that the two Lorenz-Stenflo hyperchaotic systems are projective synchronized functions.

#### Acknowledgments

This work is supported by the NSF of China (no. 11171355), the Ph.D. Programs Foundation of Ministry of Education of China (no. 20100171110040), and Guangdong Provincial NSF of China (no. S2012010010069), the Shaoguan Science and Technology Foundation (no. 313140546), and Science Foundation of Shaoguan University. The authors thank the handling editor and the reviewers for their valuable comments and suggestions, which improved the completeness of the paper.

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