Lag, Anticipated, and Complete Synchronization and Cascade Control in the Dynamical Systems

We obtain the lag, anticipated, and complete hybrid projective synchronization control (LACHPS) of dynamical systems to study the chaotic attractors and control problem of the chaotic systems. For illustration, we take the Colpitts oscillators as an example to achieve the analytical expressions of control functions. Numerical simulations are used to show the effectiveness of the proposed technique.

At the same time, many different types of synchronization in chaotic (hyperchaotic) systems were presented, for example, complete synchronization, generalized synchronization, phase synchronization, antisynchronization, general projective synchronization, lag synchronization, and anticipate synchronization, and so on.
To two dynamical systems, consider a full state hybrid projective synchronization (FSHPS) method [21], where the responses of the synchronized dynamical states synchronize up to a constant scaling matrix.In this paper, based on the Lyapunov stability theory, we propose a scheme of lag, anticipated, and complete hybrid projective synchronization control (LACHPS).In this method, every state variable of master system synchronizes other incompatible state variables of slave system; particularly, for oscillators, two different designs are shown.
When () and () are the state vectors of two dimensional chaotic systems.These two systems are completely synchronized [22] if the synchronization error ‖() − ()‖ → 0 as  → ∞.AS [23] is defined when the error ‖() + ()‖ → 0 as  → ∞.PS [24] is a situation in which the state vectors () and () synchronize up to a constant factor  (i.e.‖() − ()‖ → 0 as  → ∞.MPS [25] is defined if the state vectors of two systems synchronize up to a constant scaling matrix which means that ‖() − ()‖ → 0 as  → ∞.LS [13] implies that the state variables of the two coupled chaotic systems become synchronized but with a time lag with respect to each other; that is, ‖() − ( − )‖ → 0 as  → ∞, where  is the positive time lag.PLS has been introduced recently in [15,[26][27][28] as ‖() − ( − )‖ → 0 as  → ∞, where  is a constant scaling factor.Synchronization can be addressed as a stabilization problem.This means that the trajectories of the synchronization error have to be stabilized at the origin.
In realistic and engineering applications, LS and PLS always affect the dynamical behaviors of chaotic systems.For example, in the telephone communication system, the voice one hears on the receiver side at time  is the voice from the transmitter side at time .LS and PLS have been recently studied on systems described in [15,[28][29][30].For more details about chaotic control see [31][32][33][34][35][36][37][38] and for the elements of the cyclicity theory of planar systems see [39][40][41].Our goal in It is said that the master system and slave system are globally (i) lag hybrid projective synchronization control ( > 0,  is called the synchronization lag); (ii) hybrid complete projective synchronization control ( = 0); and (iii) anticipated hybrid projective synchronization control ( < 0, − > 0 is called the synchronization anticipation).
We remark that the above-mentioned types of synchronization are special cases of our definition.Table 1 illustrates these types of synchronization.
In order to show the results of LACHPS of two nonlinear systems, we choose the chaotic Colpitts oscillators as an example.
This paper is organized as follows.In Section 2, we show the general scheme description and theorem.In Sections 3 and 4, the Colpitts oscillator as a example is shown via applications of the LACHPS control method and cascade method.And numerical simulations are used to show the effectiveness.Finally, conclusions are drawn.
For a feasible control, the feedback  must be selected such that all the eigenvalues of (Φ + ), if any, have negative real parts.Thus, if the matrix (Φ + ) is in full rank, the system ė is asymptotically stable at the origin, which implies that ( 1) and ( 2) are in the state of LACHPS control.
Proposition 3. Let a constant matrix  and a diagonal function matrix  = diag( 1 ,  2 , . . .,   ); lag, anticipated, and complete hybrid projective synchronization between the two systems (1) and (2) will occur, if the following conditions are satisfied: (i) where , and  ∈  × ; (ii) the real parts of all the eigenvalues of (Φ + ) are negative.
Similar to the way of Theorem 5, the proof of Proposition 3 is straightforward in Appendix.
Similar to the way of the Proposition 2, the proof of Proposition 4 is straightforward in Appendix.
In order to choose a suitable control law  or a vector function , and ė () = (Φ + )() is asymptotically stable, we give the following theorem such that systems (1) and ( 2) are in the state of LACHPS control.
If  is one of the eigenvalues of matrix Φ +  and the corresponding nonzero eigenvector is , Multiplying the above equation left by   , we obtain Similarly, we also can derive that From the above two equations, we can obtain Since   [(Φ + ) + (Φ + )  ] = −, and  and  are real symmetric positive definite matrix, According to the stability theory, the system ė is asymptotically stable at the origin.

Applications of the LACHPS Control Method
Now, we introduce the following nonlinear system: where , , , and  are real constants, if  = −2,  = 0.4,  = 1.62, and  = 3, the simulation results of system (18) with the initial conditions (0, 0, 0).System (18) has chaotic attractor as shown in Figures 1 and 2. System (18) temporal evolution of the state variables is shown in Figure 3.For more detailed dynamical properties of system (2), the reader should refer to [50].
In the following, we rewrite the chaotic system (18) as a master system: and the system related to (20), 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 from ( 18) and ( 19), we obtain the error system To the LACHPS synchronization control between systems ( 18) and ( 19), we have the following theorem.Proposition 7.For the chaotic Colpitts oscillator (18), if one of the following feedback controllers   ( = 1, 2, 3) is chosen for the slave system (19) where  < 0,  > 0,  > 0,  > 0, and   and   are real, then the zero solution of the error system (21) is globally stable, and  thus (i) globally lag synchronization for  < 0, (ii) anticipated synchronization for  > 0, and (iii) complete synchronization for  = 0 occur between the master system (18) and the slave system (19).
Proof.Similar to the way of Propositions 2-4, the proof of Proposition 7 is straightforward and we omit the detail steps.We give another proof method via Lyapunov function in the following.

Remark 8. (1)
The nonlinear feedback controllers can be used to simultaneously obtain (i) hybrid lag synchronization for  > 0, (ii) hybrid anticipated synchronization for  < 0, and (iii) hybrid complete synchronization for  = 0 between the master system (19) and the slave system (20).
(2) Although the above-obtained feedback controllers are nonlinear, they are simpler than those of the so-called natural control controllers, which are derived by using with a simple stable matrix  and  for the master system (19) and the slave system (20).In the following, we obtain the numerical simulations results to prove the effective control.Numerical simulations results are presented to demonstrate the effectiveness of the proposed synchronization methods.The parameters are chosen to be [, , , ] = [−2, 0.4, 1.62, 3] in all simulations so that the chaotic system exhibits a chaotic behavior if no control is applied.The initial value [  (0),   (0),   (0)] is taken as the random number [0, 0, 0] and [  (0),   (0),   (0)] = [3,4,5].The parameters Case 1. Hybrid complete projective control: in the case  = 0, without loss of generality, the initial values of the error dynamical system (21) are  1 (0) =  1   (0) −  1   (0) = −6,  2 (0) =  2   (0)− 2   (0) = 8, and  3 (0) =  3   (0)− 3   (0) = −2.5.The dynamics of hybrid complete control errors for the master system (19) and the slave system (20) is displayed in Figures 4, 5, and 6. Figure 4 shows the chaotic attractors of the master and slave systems with different initial values in the same coordinate.Figures 5(a 3) = 0.01072905365.For simplification, we only give the dynamics of the evolutions of hybrid lag control errors for the master system (19) and the slave system (20) as displayed in Figure 8.

Applications of the LACHPS Control Method via Cascade Control Idea
In the section, based on the idea of cascade approach [42,50,51], we achieve the effectiveness control idea.Firstly, we take the system (18) as master system.The slave system is given by where ( 1 ,  2 )  is external control functions that is to be designed below.
Let the error states functions of systems ( 18) and ( 29) as follows: where  1 =  11 () +  12 ,  2 =  21 () +  22 , and  is the time lag or anticipated.The goal of the control is to find a controller ( 1 ,  2 )  such that the states of the master system (18) and the states of the slave system (29) are globally synchronized asymptotically; that is, lim Let us define the Lyapunov functions as If the Lyapunov function (32) satisfies the conditions then ( 18) and ( 29) will be satisfied.Next we take (29) as the master system, and the slave one is as follows: where ( 3 ,  4 )  is a desired controller.The relevant Lyapunov function can be chosen as where which make  3 and  4 approach to zero when  → +∞.

Conclusion
In this paper, based on the stability theory and an active control technique, we investigate the lag, anticipated, and complete hybrid projective synchronization control (LACHPS) for nonlinear chaotic systems.A nonlinear controller has been proposed to achieve lag, anticipated, and complete projective synchronization of chaotic systems.The proposed synchronization is simple and theoretically rigorous.Colpitts oscillators are used to illustrate the effectiveness of the proposed synchronization scheme.It should be note that lag synchronization control, anticipated synchronization control, and complete synchronization control.Therefore, the results of this paper are more applicable and representative.

Figure 1 :
Figure 1: Chaotic attractors for the Colpitts system with temporal evolution in different 3D spaces.

Figure 2 : 6 MathematicalFigure 3 :
Figure 2: The phase figure for the Colpitts system with temporal evolution in different plane.

Figure 6 :Figure 7 :Figure 8 :
Figure 6: The solutions of the master and slave systems with control law.(a) Signals   (the dashed line) and   (the solid line).(b) Signals   (the dashed line) and   (the solid line).(c) Signals   (the dashed line) and   (the solid line).