Generalized Synchronization of Different Chaotic Systems Based on Nonnegative Off-Diagonal Structure

The generalized synchronization problem is studied in this paper for different chaotic systems with the aid of the direct design method. Based on Lyapunov stability theory and matrix theory, some sufficient conditions guaranteeing the stability of a nonlinear system with nonnegative off-diagonal structure are obtained. Then the control scheme is designed from the stable system by the direct design method. Finally, two numerical simulations are provided to verify the effectiveness and feasibility of the proposed method.

Generalized synchronization can be regarded as an extension of complete synchronization, antisynchronization, or projective synchronization and it means that there exists a functional relation between the states of two systems.It is noted that the two systems may have different dimensions.The generalized synchronization problem between two complex networks with nonlinear coupling and time-varying delay was developed in [18].Two variants of generalized synchronization between strictly different chaotic systems were proposed in [19].Reference [20] presented a new scheme to achieve generalized synchronization between different discrete-time chaotic (hyperchaotic) systems.Reference [21] investigated the problem of chaos synchronization between two different uncertain chaotic systems with input nonlinearities.Acharyyaa and Amritkarb have briefly discussed some recent developments in the study of generalized synchronization in [22].
Recently, the direct design method [3,10] is proposed to investigate synchronization of chaotic systems.It presents an easy procedure for selecting proper controllers and the simple controller deriving from this method is easy to implement.The idea of this method is to transform the original error system into a stable system under a control scheme.It is clear that the structure of stable system plays a key role in this method.Liu et al. introduced the direct design method and realized synchronization of chaotic systems based on the stable system with tridiagonal structure [3].Subsequently, Cai et al. studied generalized projective synchronization of chaotic systems based on the stable system with antisymmetric structure [10].In this paper, the direct design method is developed to address generalized synchronization of different chaotic systems.The main contribution is to design a nonlinear control scheme by means of matrix theory.The error dynamic system is changed into a stable system with a special structure which is different from the existing results in [3,10].Here, the system  is driving the system .
The rest of this paper is organized as follows.In Section 2, the problem is formulated under investigation.The main result is given in Section 3. The stable system with nonnegative off-diagonal structure is discussed and a control scheme is presented by the direct design method.In Section 4, numerical simulations are shown to illustrate and verify the effectiveness of proposed theoretical results.The paper is concluded in the last section.
The following notations are used throughout this paper.  denotes the  dimensional Euclidean space. × is the set of all  ×  real matrices.For a vector  = [ 1 ,  2 , . . .,   ]  ∈   ,   represents the transpose of the vector , ‖‖ = (  ) 1/2 , and  > 0 if all entries   are positive.(•) is the spectral radius of matrix •.

Problem Description and Preliminaries
In this section, generalized synchronization between two coupled chaotic systems is introduced.The concept of generalized synchronization was first proposed for unidirectionally coupled systems, that is, systems coupled in a drive response configuration, by Rulkov et al. [14] (see Figure 1).Consider a system in the form of where  = [ 1 ,  2 , . . .,   ]  ∈   is the state vector, (⋅) :   →   is a continuous function.Equation ( 1) is regarded as a drive system.Then the controlled response system is described by where  = [ 1 ,  2 , . . .,   ]  ∈   ,  ∈  × is a state matrix, (⋅) :   →   is a nonlinear continuous function, and  is referred to a control scheme to be designed later.
Define the error variable  = −(), where  is continuously differentiable; then the dynamical error system is written as where () is the Jacobian matrix of the map (); that is, In fact, system (1) achieving generalized synchronization with system (2) is equivalent to the asymptotical stability of the error system (4) at  = 0. Hence, the essential problem here is how to choose a control scheme  to make system (4) asymptotically stable at the origin.
Some necessary results [23,24] in matrix theory are needed to derive our main results.

Lemma 2. Matrix 𝑄 is a nonsingular M-matrix under one of the following conditions, and the conditions are all equivalent:
(i) the eigenvalues of  have positive real parts; (ii) there exists a positive diagonal matrix  such that  +   is strictly diagonally dominant and hence also positive definite.
Lemma 3.For any matrix  ∈ Z, the following statements are equivalent to each other: (i)  is a singular M-matrix with "Property c"; (ii) there exists a symmetric positive definite matrix  such that +   is positive semidefinite; that is, matrix − is Lyapunov stable.
Lemma 4. If  ∈ Z and if there exists vector Γ > 0 such that Γ ≥ 0, then  is a singular M-matrix with "Property c. "

Main Results
In this section, we focus on the stability of a nonlinear system with nonnegative off-diagonal structure.Then a control scheme is proposed by the direct design method.Consider the following system with state dependent coefficients described by where  = [ 1 ,  2 , . . .,   ]  ∈   is the state vector, and () = [  ()] ∈  × , ,  = 1, 2, . . ., , is the coefficient matrix.
Proof.Construct a Lyapunov function where  2 is a symmetric positive definite matrix.Then the time derivative of () is If  2  +    2 is negative semidefinite and the invariant set of system (6) only includes the origin, the equilibrium  = 0 of system ( 6) is asymptotically stable.It follows from conditions (i) and (ii) and Lemma 4 that − is a singular -matrix with "Property c. " Lemma 3 guarantees the existence of matrix  2 .
Then the proof of Theorem 6 is completed.
Our main purpose is to design a control scheme  in (2) by the direct design method.The system (4) with this control scheme can be transformed into the form of system (6), that is, ė = , which is asymptotically stable at the origin according to the aforementioned conclusions.The result can be presented as follows.
Remark 8.The results obtained in the above discussion can be applied to complete synchronization [3], antisynchronization, and generalized projective synchronization [10] of chaotic systems, since complete synchronization and generalized projective synchronization are special cases of generalized synchronization.
Remark 9.It is worth noticing that the structure of matrix () in this paper is different from the existing results in [3,10].So in this sense, compared with the work of [3,10], the results of this paper will give us more freedom in designing the controller.
Remark 10.The control scheme ( 11) is designed from the stable system by the direct design method.It is worth noting that the direct design approaches transform the original systems into the system with stable tridiagonal structure by inputs.The direct design method can design the controllers with simpler structure than backstepping for nonlinear systems with low dimension.

Numerical Examples
In this section, to illustrate the effectiveness of the proposed method, we discuss two numerical simulations for two cases: (i) for identical systems, generalized projective synchronization between two Chua's circuits; (ii) for different dimensional systems, generalized synchronization between Hyper-Rössler system and Chen system.

Generalized Projective Synchronization of Chua's Circuit.
Consider the following Chua's circuit as the drive system and the controlled response system is given in the following form: where  = [ 1 ,  2 ,  3 ]  ,  = [ 1 ,  2 ,  3 ]  are the state variables,  = 10, ℎ = 16, and  = −0.143.Generalized projective synchronization of systems ( 13) and ( 14) will be realized with () = , where  is a nonzero constant and referred to as a scaling factor.Defining the error Figure 2: Antisynchronization errors and state trajectories of systems ( 13) and ( 14) under the controller (16).

Generalized Synchronization between HyperRössler System and Chen System. Consider the HyperRössler system
as drive system and the controlled response system is the following Chen system: where  = [ 1 , Defining the error variable  = −(), it follows from ( 22)- Design the controller It is clear that all conditions of Theorem 5 and Theorem 6 are satisfied for system (25), where condition (ii) of Theorem 6 can be verified by choosing Γ = [1,1,1]  .According to Theorem 5 or Theorem 6, system (25) is asymptotically stable at the origin.Hence, system (22) achieves generalized synchronization with system (23) under the controller (26).Chaotic attractors of HyperRössler system and Chen system with initial states (0) = [−10, 1, 2, 0]  and (0) = [3,8,10]  are shown in Figures 4 and 5, respectively.Numerical simulation for generalized synchronization of systems ( 22) and ( 23) under controller (26) is shown in Figure 6.

Conclusions
This paper has studied generalized synchronization of different chaotic systems.By using Lyapunov stability theory and matrix theory, the stability of a nonlinear system with nonnegative off-diagonal structure is obtained.Then the control scheme can be designed from transforming the error system into this stable system by using the direct design method.It is worth noticing that the structure of this stable system is significantly different from existing results of the direct design method.Numerical simulations are presented for identical systems and nonidentical systems with different dimension, respectively.

Figure 1 :
Figure 1: Schematic diagram of a drive-response configuration.Here, the system  is driving the system .