## Stability and Bifurcation Analysis of Differential Equations and its Applications

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# Stabilization and Synchronization of Unified Chaotic System via Impulsive Control

**Academic Editor:**Yonghui Xia

#### Abstract

The impulsive control and synchronization of unified chaotic system are proposed. By applying impulsive control theory and introducing a piecewise continuous auxiliary function, some novel and useful conditions are provided to guarantee the globally asymptotical stabilization and synchronization of unified chaotic system under impulsive control. Compared with some previous results, our criteria are superior and less conservative. Finally, the effectiveness of theoretical results is shown through numerical simulations.

#### 1. Introduction

In 1963, Lorenz presented the first chaotic attractor in a simple three-dimensional autonomous system [1]. Since then, many researchers began to study chaos theory. In 1999, Chen and Ueta found another chaotic system called Chen system, which is similar but topologically nonequivalent to Lorenz system [2]. In 2002, the unified chaotic system was introduced by Lü and Chen. It includes Lorenz system and Chen attractor as two extremes, respectively, and Lü attractor as a transition system between Lorenz system and Chen system [3, 4].

For a long time, people thought that chaos was neither predictable nor controllable. However, in the 1990s, the viewpoint and theory about control of chaotic system introduced by Ott. Grebogi Yorke and Pecora Carroll completely changed the situation [5], and many scholars began to study chaos control. Until now, many different techniques and methods have been proposed to achieve chaos control and synchronization, such as adaptive control [6], fuzzy control [7], and impulsive control [1, 5, 8–22].

In the past several years, impulsive control strategy has been widely used to stabilize and synchronize nonlinear dynamical systems. From the control point of view, impulsive control is an effective method based on the theory of impulsive dynamic systems [14, 17, 23]. It allows stabilization of a complex system by using only small control impulses. Additionally, this method drastically reduces the amount of information transmitted from the driving system to the driven system in chaos synchronization [10, 20]. Those merits make it more efficient and more useful in a great number of real-life applications. Up to now, the stabilization and synchronization of chaotic systems under impulsive control have been extensively studied (see [1, 5, 8–10, 15, 16, 19–22]).

In this paper, based on impulsive control theory, the stabilization and synchronization of unified chaotic system are investigated via designing impulsive controller. By introducing a piecewise continuous function, some new and useful criteria are established to guarantee the stabilization and synchronization of unified chaotic system. Compared with the main criteria obtained in [8, 10], our results are less stringent and more general.

This paper is organized as follows. In Section 2, two central lemmas are given. In Section 3, the impulsive control model of unified chaotic system is introduced and some new conditions are provided to ensure the stability of the equilibrium point for unified chaotic system under impulsive control. In Section 4, impulsive synchronization criteria are proposed. In Section 5, numerical simulations are provided to illustrate and verify our results.

#### 2. Preliminaries

In this section, two central lemmas are provided to support our analysis in later sections.

Lemma 1. *Assume that is an symmetric and semipositive definite real matrix; then, for any two real column vectors and , the following inequality holds:
*

*Proof. *Since matrix is symmetric and semipositive definite, there exists an matrix such that
Then, by using Cauchy-Bunyakovsky inequality, we have
It follows that
This proof is completed.

Lemma 2. *Assume that is an symmetric real matrix and and are the smallest and the largest eigenvalues of , respectively. Then, for an arbitrary column vector , the following inequality holds:
*

*Proof. * is a symmetric real matrix; then, there exists an orthogonal matrix such that
where and is the th eigenvalue of , .

For an arbitrary column vector , we have

Similarly, we can prove

The proof is completed.

#### 3. Impulsive Control of Unified Chaotic System

Consider the unified chaotic system [3, 4, 8, 10] described by the following form: where are state variables and . System (9) has some special features and advantages because it unifies Lorenz system (when ), Lü system , and Chen system (when ). Moreover, system (9) is always chaotic in the whole interval .

It is easy to see that system (9) has three equilibrium points: where and .

For convenience, denote the above equilibrium points as and introduce the following transformation: Then, from (9), the following model can be obtained: We decompose the linear and nonlinear parts of system (12) and rewrite it as the following form: where and

In order to stabilize the equilibrium point, we now introduce a control model of system (13) described by where is the Dirac delta function, the time sequence satisfies , , and denotes a constant matrix for each .

Integrating from to both sides of system (15), we have where is sufficiently small. As , by applying the properties of the Dirac delta function, we have Throughout this paper, we always assume that is left continuous at (); that is, . In this case, the above equality is equivalent to the following form: Thus, from the definition of the Dirac delta function, control system (15) can be rewritten as It is easy to see that the origin is an equilibrium point of system (19). We are interested in stabilizing the origin of system (19), which is equivalent to the asymptotical stability of the equilibrium point of system (9).

For convenience, denote , () and assume that Since system (9) is chaotic, there exist two positive numbers and such that , , and for (see [4]). Additionally, we have , , and for .

The following theorem is provided to ensure the globally asymptotical stability of the origin for the impulsive control system (19).

Theorem 3. *Let be a symmetric and positive definite matrix such that is semipositive definite, where denotes the identity matrix. Denote that and are the smallest and the largest eigenvalues of and , respectively. Let be the largest eigenvalue of . is the largest eigenvalue of the matrix and , . Denote that
**
The origin of the controlled unified chaotic system (19) is globally asymptotically stable if there exist two constants and such that one of the following conditions is satisfied.*(1)*For all ,
*(2)*For all ,
*

*Proof. *Firstly, construct an auxiliary function described by the following form:
Denote that . For , if , in view of Lemmas 1 and 2, the derivative of along the solution of (19) is
On the other hand, if ,
From what has been discussed above, we have
Besides, for ,
From (27), for ,
which leads to
Further, by using (28),
Similarly, for , we obtain
Hence, generally, for ,
If , then
If , then
where
Hence, for , we always have

In the following, we consider the following cases.*Case 1*. When condition (9) is satisfied, from (22),
and together with (37), we have
Note that as ; then, from , we obtain . Therefore, we finally have .*Case 2*. If condition (12) holds, from (23), we have
For , it follows from (37) and (40) that
Similarly, if , from (37) and (40), we can derive that
From (41) and (42), we finally have .

From the above analysis, the trivial solution of system (19) is globally asymptotically stable. The proof of Theorem 3 is completed.

*Remark 4. *From Theorem 3, it is clear that if . On the other hand, for , as approaches to , since in this case.

*Remark 5. *If in Theorem 3, from the proof of Theorem 3, we have
where . In this case, it is clear that the origin of impulsive control system (19) is stable.

Taking in Theorem 3, it is easy to see that , , and in this case. By virtue of Theorem 3, the following statements can be derived.

Corollary 6. *The origin of the controlled unified chaotic system (19) is globally asymptotically stable if there exist two constants and such that one of the following conditions holds.*(a)*For all ,
*(b)*For all ,
*

*Remark 7. *In Corollary 6, if , inequality (44) is reduced to
which is equivalent to the condition of Theorem 1 in [10]. Further, in this case, inequality (45) is degenerated to
which is the same as inequality in [8]. On the other hand, for , evidently, conditions (44) and (45) are less conservative than inequalities (46) and (47). Hence, compared with [8, 10], our results are less stringent and more general.

In practice, the equal impulsive interval and the constant gain matrix are often selected for convenience. In Corollary 6, we choose and for and denote that and are the largest eigenvalues of and , respectively. Then, the following corollary can be obtained.

Corollary 8. *The origin of system (19) is globally asymptotically stable if there exist and such that
*

*Remark 9. *From Corollary 8, an upper bound of equal impulsive interval can be derived. In fact,
In [10], the upper bound of impulsive interval was also given, which can be represented by
It is evident that the equal impulsive interval obtained in this paper is larger, which shows that our designed controller is more efficient and more economical in practice.

#### 4. Impulsive Synchronization of Unified Chaotic System

In this section, we discuss the impulsive synchronization of unified chaotic system. It is easy to see that system (9) can be rewritten as where and In impulsive synchronization configuration, the driving system is given by (51), while the driven system is represented by the following form: where is the state variable of the driven system, and are defined in (52), and , , and are defined in (15).

Similar to the technique of system (15), the driven system (53) can be rewritten as the following impulsive equation: Let be the synchronization error and From (51) and (54), the error system can be expressed by

In this section, we are interested in synchronizing the driving system (51) and the driven system (54), which is equivalent to the asymptotical stability of the origin for error system (56). Similar to Theorem 3, the following results are stated, which ensure the globally asymptotical synchronization of the driving system (51) and the driven system (54).

Theorem 10. *Let be the largest eigenvalue of . is the largest eigenvalue of the matrix . The driving system (51) and the driven system (54) are globally asymptotically synchronized if there exist constants and such that one of the following conditions is satisfied.*(a)*For all ,
*(b)*For all ,
*

*Proof. *Observe that error system (56) is almost the same as system (19) except for . Construct an auxiliary function described by the following form:
For , calculating the time derivative of along the solution of (56), we have
In addition, for ,
The rest of this proof is the same as that of Theorem 3, and here we omit it. The proof of Theorem 10 is completed.

Similar to Corollary 6, we have the following results.

Corollary 11. *The driving system (51) and the driven system (54) are globally asymptotically synchronized, if there exist and such that one of the following conditions is satisfied.*(a)*For all ,
*(b)*For all ,
*

*Remark 12. *The impulsive synchronization of unified chaotic system was investigated in [8, 10]. It is noted that inequality (62) is reduced to the condition of Theorem 2 in [10] and condition (b) in Corollary 11 will be degenerated to the assumption of Theorem 2 in [8] when . Additionally, if , it is evident that conditions (62) and (63) are less stringent than the assumptions introduced in [8, 10].

Similar to Corollary 8, the following results can be derived when we select and for each .

Corollary 13. *The driving system (51) and the driven system (54) are globally asymptotically synchronized, if there exist and such that
**
where is the largest eigenvalue of .*

#### 5. Numerical Simulations

In this section, based on the results obtained in the previous sections, some numerical simulations are represented to show the effectiveness of our results.

First, we consider numerical simulations of impulsive control for system (9) with . The chaotic behavior of system (9) with is shown in Figure 1.

For convenience, select (, and take gain matrices in Theorem 3. By simple computation, we obtain and its eigenvalues are , , and ; then, .

Choosing impulsive control interval , . By computation, and Taking in Theorem 3, then, Hence, Therefore, all conditions of Corollary 8 are satisfied, which implies that the origin of unified chaotic system (9) is globally asymptotically stable under impulsive control. The numerical simulation results with constant impulsive interval are shown in Figure 2.

In the second simulation, we study impulsive synchronization of the driving system (51) and the driven system (54) with . The initial conditions of the driving system and the driven system are and , respectively. Figure 3 shows that system (51) with is chaotic, and from this we can obtain that .

Choose , , , and in (43). By computation, , and and , which leads to By the above analysis, all conditions of Corollary 13 hold; hence, system (51) and system (54) are globally asymptotically synchronized under impulsive control. The synchronization errors with impulsive distance are shown in Figure 4.

*Remark 14. *Evidently, it follows from (66) and (69) that the conditions of Theorems 1 and 2 in [10] are not satisfied under the above parameters; that is to say, the stabilization and synchronization of unified chaotic system are not ensured by means of the criteria obtained in [10]. However, it follows from (68) and (70) that the stabilization and synchronization of unified chaotic system can be realized by virtue of our criteria and Figures 2 and 4 show the validity of our results. Therefore, our results are less conservative and more superior compared with [10].

#### 6. Conclusion

In this paper, the issue on the stabilization and synchronization of unified chaotic system under impulsive control is investigated. Some novel and useful criteria are derived by using impulsive control theory. Finally, the effectiveness and feasibility of the developed methods are shown by some numerical simulations. Compared with some previous results given in [8, 10], our designed controller is more efficient and economical.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the Natural Science Foundation of Xinjiang (Grant no. 2013211B06), the Project funded by China Postdoctoral Science Foundation (Grant no. 2013M540782), the Natural Science Foundation of Xinjiang University (Grant no. BS120101), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20136501120001), and the National Natural Science Foundation of China (Grant no. 61164004).

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#### Copyright

Copyright © 2014 Cheng Hu and Haijun Jiang. 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.