Stabilization and Synchronization of Unified Chaotic System via Impulsive Control

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.


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].
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.

Preliminaries
In this section, two central lemmas are provided to support our analysis in later sections.
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 Λ = diag( 1 ,  2 , . . .,   ) and   is the th eigenvalue of ,  = 1, 2, . . ., .For an arbitrary column vector  ∈   , we have Similarly, we can prove The proof is completed.
For convenience, denote the above equilibrium points as ( 0 1 ,  0 2 ,  0 3 )  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  = ( 1 ,  2 ,  3 )  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 and   denotes a 3 × 3 constant matrix for each  ∈  + = {1, 2, . . ., ,  + 1, . ..}.
Integrating from   − ℎ to   + ℎ both sides of system (15), we have where ℎ > 0 is sufficiently small.As ℎ → 0 + , by applying the properties of the Dirac delta function, we have Throughout this paper, we always assume that 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 ( 0 1 ,  0 2 ,  0 3 )  of system (9).For convenience, denote   =  +1 −   , Δ  =  2+1 −  2−1 ( ∈  + ) and assume that sup Since system ( 9) is chaotic, there exist two positive numbers 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 3 × 3 symmetric and positive definite matrix such that (−) is semipositive definite, where  denotes the identity matrix.Denote that  1 and  2 are the smallest and the largest eigenvalues of  and (−), respectively.Let  be the largest eigenvalue of ( +   ).  is the largest eigenvalue of the matrix ( +   )  ( +   ) and The origin of the controlled unified chaotic system (19) is globally asymptotically stable if there exist two constants  > 1 and  ≥ 0 such that one of the following conditions is satisfied.
From the above analysis, the trivial solution () = 0 of system (19) is globally asymptotically stable.The proof of Theorem 3 is completed.
Remark 4. From Theorem 3, it is clear that  = 0 if  = .On the other hand, for  ̸ = ,  → 0 as  approaches to , since  2 → 0 in this case.
Taking  =  in Theorem 3, it is easy to see that  1 = 1,  2 = 0, and  = 0 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  > 1 and  ≥ 0 such that one of the following conditions holds.
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 (7) in [8].On the other hand, for  > 0, 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.(19) is globally asymptotically stable if there exist  > 1 and  ≥ 0 such that ( − )  + ln () ≤ 0.

Corollary 8. The origin of system
(48) 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.

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  = ( 1 ,  2 ,  3 )  and  = ( − (25 + 10) 25 + 10 0 28 − 35 29 − 1 0 In impulsive synchronization configuration, the driving system is given by (51), while the driven system is represented by the following form: where ỹ = (ỹ 1 , ỹ2 , ỹ3 )  is the state variable of the driven system,  and Φ 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  = ( 1 ,  2 ,  3 )  = (ỹ 1 −  1 , ỹ2 −  2 , ỹ3 −  3 )  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 ( +   ) T ( +   ).The driving system (51) and the driven system (54) are globally asymptotically synchronized if there exist constants  > 1 and  ≥ 0 such that one of the following conditions is satisfied.
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  > 1 and  ≥ 0 such that one of the following conditions is satisfied.
(a) 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  = 0. Additionally, if  > 0, 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  > 1 and  ≥ 0 such that where  is the largest eigenvalue of ( + )  ( + ).

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  = 1.The chaotic behavior of system (9) with  = 1 is shown in Figure 1.
In the second simulation, we study impulsive synchronization of the driving system (51) and the driven system (54) with  = 0.The initial conditions of the driving system and the driven system are (−2, −8, −1) and (5, −9, −6), respectively.Figure 3 shows that system (51) with  = 0 is chaotic, and from this we can obtain that  2 = 65.
Choose 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].

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.
Assume that  is an  ×  symmetric and semipositive definite real matrix; then, for any two real column vectors  ∈  n and  ∈   , the following inequality holds: 2 ≤              ⋅              .