Synchronization and Antisynchronization for a Class of Chaotic Systems by a Simple Adaptive Controller

This paper investigates the synchronization and antisynchronization for a class of chaotic system. Firstly, a necessary and sufficient condition is proposed to synchronize and antisynchronization simultaneously for the chaotic systems. Secondly, two methods are obtained to realize coexistence of synchronization and antisynchronization in the chaotic systems, and the corresponding adaptive controllers are also given. Finally, two numerical examples with simulation verify the correctness and effectiveness of the obtained results.


Introduction
Since Lorenz firstly found the classical chaotic attractor in 1963 [1], as a most fascinating phenomenon in nonlinear dynamical system, chaos has been intensively studied over the past few decades; see [2,3] and the references therein.It is well known that Pecora and Carroll firstly investigated the synchronization problem of chaotic systems in 1990 [4], and Ott et al. firstly presented a method to control chaotic systems successfully in 1990 in [5].From then on, chaos control and chaos synchronization have received a great deal of attention in the area of nonlinear control as the significance of these two problems in both academic research and practical applications, and many important results were obtained; please refer to [6][7][8][9][10][11].
Up to date, several types of typical synchronization have been identified such as complete synchronization (CS), phase synchronization (PS), lag synchronization (LS), generalized synchronization (GS), anti-phase synchronization (AS), and projective synchronization (PS), and a variety of works have been done about the above problems; see [6,7,[12][13][14][15][16] and the references therein.It is well known that the master system synchronizes the slave system which is equivalent to the error system that is asymptotically stable.That is to say, chaos synchronization is equivalent to the error system which is asymptotically stable.Similarly, the master system antisynchronizes the slave system which is equivalent to the sum system that is also asymptotically stable.From the view of control theory [17], in order to design a simple and physical controller, the following condition is necessary; that is,  = 0 is an equilibrium point of the unforced nominal error system ė = () − (), ẋ = (), where  =  − , and  = 0 is also an equilibrium point of the unforced nominal sum system Ė = () + (), ẋ = (), where  =  + .Obviously,  = 0, that is,  = , is an equilibrium point of the error system ė = () − ().Whereas,  = 0, that is,  = −, is an equilibrium point of the error system Ė = () + () if and only if (−) = −().Thus, the antisynchronization problem is more complex than the synchronization problem.However, this necessary condition is not considered in the most of the existing works on antisynchronization of chaotic systems [14,15].Although the authors have solved the antisynchronization of chaotic systems successfully, the controllers that have been obtained are complex; that is, some terms in those controllers are needed to counteract the redundant terms which make  = 0 not the equilibrium point of the sum system Ė = () + ().For example,  2  2 +  1  1 in  2 of ( 14) counteracts the redundant term − 2  2 −  1  1 in error system (13), and − 1  1 −  2  2 in  3 of ( 14) also does; for details please see [15].
It should be pointed out that most of the existing works focus on investigating the same kind synchronization in a given chaotic system; that is, all the states of the slave system have the same kind synchronization to the corresponding states of the master system.For example, when we say that two systems are synchronized (or antisynchronized, or lagsynchronized, or something else) with each other, it means that each pair of the states between the interactive systems is complete synchronous (or antisynchronous, or something else).In [18], the authors firstly pointed out the coexistence and switching of anticipating synchronization and lag synchronization in an optical system.From then on, some important results have been obtained; see [19,20].However, there are no results which can give some conditions or algorithms to select what variables in the master chaotic system which can synchronize or antisynchronize the corresponding variables in the slave chaotic systems have been published so far.Therefore, the coexistence of synchronization and antisynchronization of a class of chaotic or hyperchaotic systems needs further research.
Motivated by the above two reasons, we investigate the synchronization and antisynchronization for a class of chaotic systems in this paper.Firstly, for a class of chaotic systems, we obtain a necessary and sufficient condition with which the master system can synchronize and antisynchronize the slave system simultaneously.Secondly, we give two methods to realize coexistence of synchronization and antisynchronization in the chaotic systems and design the corresponding adaptive controllers.Finally, two numerical examples with simulation verify the correctness and effectiveness of the obtained results.

Preliminary Knowledge
This paper studies the synchronization and antisynchronization for a class of chaotic systems by adaptive control method.In order to develop this paper, some assumption and definitions are introduced firstly.
Assumption 1 (see [17]).  = 0 is an equilibrium of the nonlinear system ẋ = (); that is, (  ) = 0. Remark 2. Assumption 1 is a basic assumption of the system control theory.Without loss of generality, if   ̸ = 0, we can obtain a new system ẏ = ( +   ) whose equilibrium is   = 0 by making a coordinate transform  =  −   .Definition 3 (see [12]).Consider the following chaotic system: where  ∈ R  is the state and () is a smooth nonlinear vector function.
Let system (1) be the master system; then the corresponding slave system with controller  is given as where  ∈ R  is the state and  = ( 1 ,  2 , . . .,   )  is the controller to be designed.Let  =  − , and the error system is described as We call master chaotic system (1) and slave system (2) reach complete synchronization if lim  → ∞ ‖()‖ = 0.
With the development of this paper, we introduce our previous result which can make the error system or the sum system reach stabilization.

Main Results
In this section, we firstly give a necessary and sufficient condition for a class of chaotic systems, by which we can determine whether master system (1) and slave system (2) realize synchronization and antisynchronization simultaneously or not.
Remark 9. Although, the problem of synchronization and antisynchronization simultaneously of 4-dimension hyperchaotic system has been investigated in [22], no sufficient or necessary and sufficient condition for the general chaotic systems was proposed.Theorem 8 gives a necessary and sufficient condition for a class of chaotic systems.
If () is not an odd function, master system (1) and slave system (2) cannot realize synchronization and antisynchronization simultaneously according to Theorem 8.Under this condition, they can reach the coexistence of synchronization and antisynchronization.Then, we give two methods to realize the coexistence of synchronization and antisynchronization for a class of chaotic systems.
According to Assumption 1, we give the following conclusion.
Remark 12. Theorem 11 gives a condition which can determine what variables in master system (1) can synchronize the corresponding variables in slave system (2), while other variables in master system (1) can antisynchronize the corresponding variables in slave system (2).

Illustrative Example
In this section, we give two numerical examples to illustrate how to use the results we obtained in this paper to realize the synchronization and antisynchronization simultaneously and the coexistence of synchronization and antisynchronization, respectively.
Obviously, system (14) satisfies Theorem 8; thus this system can realize the synchronization and antisynchronization simultaneously.
Example 16.Consider the following chaotic system [24]: The above system is called unified chaotic system, where  ∈ [0, 1].If  ∈ [0, 0.8), the system is generalized Lorenz system; if  ∈ (0.8, 1], the system is generalized Chen system.Let system (17) be the master system, and then the slave system is described as Mathematical Problems in Engineering 5 Let   =   −     , where |  | = 1,  = 1, 2, 3. Then the unforced nominal error system is described as follows: According to Assumption 1,  = 0 is the equilibrium of the above unforced nominal error system (19), and the following algebraical equations should be satisfied.
Select  1 =  2 = −1,  3 = 1, and unforced nominal sum and error system (19) is given as It is easy to obtain that if  2 = 0, then remainder sum and error system (21) is asymptotically stable.
According to Lemma 7, forced sum and error system (21) is given as that is, the controller is  = (0,  1  2 , 0)  and k 1 = − 2 2 .Next, for Example 16, we can obtain the same conclusion by using Theorem 13.
Remark 17.From the results of numerical simulation, Figure 3 shows that sum and error system ( 23) is asymptotically stable, while Figure 4 shows that master system (17) and slave (18) realize coexistence of synchronization and antisynchronization.

Conclusions
In this paper, we have investigated the synchronization and antisynchronization for a class of chaotic systems.Firstly, a necessary and sufficient condition has been proposed, with which the master system can synchronize and antisynchronize the slave system simultaneously.Secondly, two methods have been obtained to realize coexistence of synchronization

3 Figure 1 : 3 Figure 2 :
Figure 1: The response of the error system.

3 Figure 3 :
Figure 3: The response of sum and error system.

3 Figure 4 :
Figure 4: The response of master and slave system.