Synchronization of a Class of Chaotic Systems with Different Dimensions

In this paper, two scaling matrices are used to research the synchronization of different dimensional chaotic systems with unknown parameters. Firstly, the definition of synchronization of chaotic systems with different dimensions is introduced. Secondly, based on Lyapunov stability theorem and adaptive control method, an adaptive feedback hybrid controller and parameter adaptive laws are designed to realize synchronization of uncertain chaotic systems with different dimensions. Finally, three numerical experiments are carried out to verify the effectiveness of the proposed method.


Introduction
Chaos is a very complex nonlinear motion, which will show randomness, boundedness, unpredictability, and sensitive dependence on the initial value in the evolution process [1]. Chaotic system is different from the general system, and it is greatly affected by the changes in initial value and parameters, through the continuous in-depth understanding of the characteristics of chaotic motion, gradually realizing that the existence of chaos has brought some damage to the system; for example, chaotic motion can cause the circuit to form irregular oscillation and make the chaotic motion gradually away from the target trajectory, which may cause unpredictable consequences. At the same time, chaos also enlarges the influence of small changes and improves their effectiveness of achieving a specific desired state with very high flexibility and low cost. In some research fields, this property may be a valuable advantage. In 1990, Pecora and Carroll first realized chaos synchronization in electronic circuits [2], which set off an upsurge of chaos synchronization control research, making chaos synchronization widely used in many fields, such as communication encryption, engineering optimization, and statistical prediction [3][4][5][6].
In recent years, with the in-depth understanding of chaotic systems, some typical synchronization problems have been studied, such as complete synchronization, antisynchronization, projective synchronization, and hybrid synchronization [7][8][9]. However, in most of the control strategy design process, the system will be affected by various uncertain factors, and it is difficult to directly obtain some accurate mathematical models of the system. Some uncertain parameters of the chaotic system may have a certain impact on the performance of the system. On the contrary, due to the influence of these uncertain factors, the synchronization behavior of the system may be destroyed, and the emergence of unknown quantities in the system makes the synchronization control very complex. Many existing kinds of literatures consider the synchronization between two uncertain chaotic systems in the same dimension; for example, in [10], an adaptive sliding mode disturbance observer is designed to research the synchronization problem of uncertain Dadras-Momeni chaotic systems with time-varying disturbances. In [11], a finite-time synchronization scheme is proposed for uncertain chaotic systems with external disturbances, parametric uncertainties, Lipschitz nonlinearities, and time delays. In [12], an adaptive complete synchronization scheme is designed for uncertain financial chaotic systems with unknown parameters. Synchronization behavior also exists in different dimensional systems in practice, and it is also especially important; for example, the circulatory and respiratory systems with different dimensions exist in synchronization behavior [13]. Similarly, in the communication network, the different dimension structure design and synchronization control of the driving system and the response system will increase the difficulty for the listener to decipher and make the communication more secure [14]. erefore, it is of great significance to research the synchronization of chaotic systems with uncertain parameters and different dimensions in practical applications. In [15], the synchronization of a particular chaotic system with unknown parameters is studied by constructing the cross-error difference term. Recently, there have been some research studies on achieving the synchronization of chaotic systems by reducing dimension [16,17], as well as some research studies by increasing dimension and reducing dimension simultaneously [18,19]. In [20][21][22], the author further studies the synchronization of different dimensional integer-order or fractional-order chaotic or hyperchaotic systems using scaling matrices. e main contribution of this paper is to research the scaling matrix synchronization of different dimensional chaotic or hyperchaotic systems with unknown parameters. A new suitable controller and parameter adaptive laws are designed to realize the synchronization of chaotic systems with different dimensions in a certain dimension. Simultaneously, according to the characteristics of the system, a transformation matrix is introduced, which can upgrade or reduce the order of the defined error system to meet the needs of different dimensional systems. Compared with [15][16][17][18][19], the synchronization scheme presented in this paper will not destroy the characteristics of system dimension and will directly synchronize chaotic systems with different dimensions in a certain dimension; on the other hand, the scaling matrix synchronization method used in this paper is not limited to a single synchronization type and can be transformed into a variety of synchronization schemes. In [20][21][22], although the author further studies the synchronization of chaotic systems with different dimensions by using the scale matrix, most of the results are only applicable to the fully known deterministic chaotic systems. However, they cannot do anything for the uncertain chaotic systems with unknown parameters, and we have done further research on the above work and extended the synchronization problem to the uncertain chaotic or hyperchaotic systems. Because the hyperchaotic system has at least two positive Lyapunov exponents, compared with chaotic system, the higher the degree of instability, the more complex the chaotic behavior. At the same time, according to the characteristics of these systems, appropriate controllers are designed to synchronize and stabilize the chaotic and hyperchaotic systems with different dimensions, which has certain requirements for the design of controllers.
ese factors bring great difficulties to our research. is paper is organized as follows. Preliminaries are presented in Section 2. e synchronization theory of different dimension chaotic systems is introduced in detail, and the main conclusions are presented in Section 3. ree examples are presented in Section 4. Finally, in Section 5, concluding comments are given.

Preliminaries
Consider the following drive system in the form of where x ∈ R n is the state vector of the drive system and F(x): R n ⟶ R n defines a vector field in n-dimensional space. e response system is assumed by where y ∈ R m is the state vector of the response system, G(y): R m ⟶ R m defines a vector field in m-dimensional space, and u � (u 1 , . . . , u m ) T ∈ R m is the control input vector.
Definition 1 (see [20]). e drive system (1) and the response system (2) are said to be synchronized in dimension d, with respect to scaling matrices Θ and Φ, if there exists a controller u ∈ R m and given matrices Θ ∈ R d×m and Φ ∈ R d×n such that the synchronization error e � Θy − Φx satisfies that where d is a certain constant and ‖ · ‖is the vector norm.
In short, if this is the case, then the drive system (1) and the response (2) achieved Θ − Φ synchronization. Remark 1. Θ and Φ are constant scaling matrices.
Remark 2 (see [22]). It is easy to see that depending on our choice of the matrices Θ and Φ, we may have several synchronization types.

Scaling Matrix Synchronization of Chaotic Systems with Unknown Parameters
In this part, our main results include three parts: firstly, considering some unknown parameters in chaotic systems (1) and (2), we design new controller and parameter adaptive laws and give sufficient conditions to ensure the Θ − Φ synchronization of chaotic systems with different dimensions; secondly, according to the characteristics of the system model, some new sufficient conditions are given to achieve the desired results; finally, some interesting corollaries are given.
Based on [20], we consider the problem of Θ − Φ synchronization between n-dimensional drive system and mdimensional response system with unknown parameters.
It is assumed that the drive system and response system with unknown parameters can be written in the following form: where α � (α 1 , α 2 , . . . , α τ 1 ) T are parameters of drive system, β � (β 1 , β 2 , . . . , β τ 2 ) T are parameters of response system, τ 1 and τ 2 are constants, and f: Considering the error system of Definition 1, then the dynamic equation of the error system is Assumption 1. In general, 0 < d ≤ max m, n { }. For the convenience of discussion, we will consider the case d � m of dimension in this paper.

Assumption 2.
e matrix Θ is row full rank matrix.
Remark 3 (see [23]). If the matrix Θ is row full rank matrix, then define Θ − 1 as the right inverse of a matrix Θ.
Since there are unknown parameters in the systems, we assume that α and β represent the estimations of α and β, respectively. Denote where Θ ∈ R m×m and Φ ∈ R m×n are the scaling matrices and P ∈ R m×m is the positive-definite control matrix. For the convenience of calculation, let P � diag(p 1 , p 2 , . . . , p m ), p i > 0, i � 1, . . . , m.
e following adaptive laws are proposed to estimate the unknown parameters α and β: where Δα � α − α, Δβ � β − β, and ρ and ρ ′ are the constant gain matrices. For the convenience of calculation, . . , τ 2 . en, based on equation (7), the controller can be designed as follows: Hence, the corresponding error system is obtained as Theorem 1. For the chaotic systems (4) with unknown parameters, if the controller u is designed as equation (8) and the parameter adaptive laws are designed as equation (7), then the error system (9) is asymptotically stable at the origin. It can also be said that the response system and drive system in Proof. Select the Lyapunov function as follows: en, the derivative of V(t) along equations (7), (8), and (9) can be calculated as It is clear that V(t) is positive definite and _ V(t) is negative definite in the neighborhood of the zero solution for system (9). According to Lyapunov stability (see eorem 4.1 in [24]), the synchronous error system (9) is asymptotically stable at the origin. By Definition 1, the synchronization problem of chaotic systems can be transformed into the problem of asymptotic stability of synchronization error system at the origin, so the systems are uniformly asymptotically stable. erefore, under the action of controller (8) and the parameter adaptive laws (7), the drive system and the response system in (4) achieved Θ − Φ synchronization.
is completes the proof of eorem 1. We consider the expression (7) of the parameter adaptive laws in eorem 1 and transform the constant gain matrix into the function of a gain variable, then the parameter adaptive laws in eorem 1 can be written as where ρ(t): R ⟶ R is a gain function.
Complexity 3 e adaptive law of gain function is designed as follows: e following Corollary 1 can be obtained.
□ Corollary 1. For the above chaotic systems (4), keeping the controller in eorem 1 unchanged and replacing the constant gain matrix with the gain function in the parameter adaptive laws, the system can still achieve Θ − Φ synchronization under the action of the controller (8), the parameter adaptive laws (12), and the gain function adaptive law (13).
Proof. e Lyapunov function is selected as e proof method is similar to eorem 1 and omitted. Let us denote α i or β i be the ith unknown parameter of ith equation in the system. Assume that system (4) is linear at unknown parameters α i or β i . en, systems (4) can be written as where A α and B β are the diagonal matrix composed of unknown parameters α and β, where e � (e 1 , . . . , e m ) T , e � (e 1 , . . . , e n ) T , and Ω � (Ω ij ) ∈ R n×m is the transformation matrix. If n ≤ m, where e following new adaptive laws are proposed to estimate the unknown parameters α and β: where ψ and ψ ′ are the constant gain matrices. For the convenience of calculation, Combined with the adaptive laws of the parameter, the following controller is designed: By substituting the controller (19) into the error system (5), it can be obtained: □ Theorem 2. For the chaotic system (15) with unknown parameters, if the controller v is designed as equation (19) and the parameter adaptive laws are designed as equation (18), then the error system (20) is asymptotically stable at the origin. It can also be said that the response system and drive system in (15) Proof. Select the Lyapunov function as follows: en, the derivative of V(t) along equations (18), (19), and (20) can be calculated as Similar to the proof method of eorem 1, it is easy to obtain that the error system (20) is asymptotically stable at the origin. By Definition 1, the synchronization problem of chaotic systems can be transformed into the problem of asymptotic stability of synchronization error system at the origin. erefore, under the joint action of the controller (19) and the parameter adaptive laws (18), the drive system and the response system in (15) is completes the proof of eorem 2. e parameter adaptive laws (18) in eorem 2 is considered, and the constant gain matrix is transformed into the function of gain variable, and then, the parameter adaptive laws in eorem 1 can be written as where ψ(t): R ⟶ R is a gain function. e adaptive law of gain function is designed as follows: □ Corollary 2. For the above chaotic systems (15), if the controller in eorem 2 remains unchanged and the constant gain matrix is replaced by the gain variable function in the parameter adaptive laws, the systems can still achieve Θ − Φ

Complexity
synchronization under the joint action of the controller (19), the parameter adaptive laws (23), and the gain function adaptive law (24).
Proof. e Lyapunov function is selected as e proof method is similar to eorem 2 and omitted.

Numerical Simulations
In this section, we give three examples to verify the effectiveness of our proposed method. Example 1. Consider the chaotic Cai system [25] as the drive system and the hyperchaotic Rossler system [26] as the response system. e system equations are as follows: where x 1 , x 2 , x 3 are the drive system state variables; y 1 , y 2 , y 3 , y 4 are the response system state variables; u 1 , u 2 , u 3 , u 4 are the designed controllers, when the parameters are taken as a 1 � 20, b 1 � 14, c 1 � 10.6, d 1 � 2.8 and a 2 � 0.25, b 2 � 3, c 2 � 0.5, d 2 � 0.05; and the systems (26) and (27) are chaotic and hyperchaotic. e projection of attractor of chaotic Cai system and hyperchaotic Rossler system is shown in Figures 1 and 2, respectively.
Let make the drive system and the response system achieve synchronization in 4 D. We rewrite the systems (26) and (27) in the following forms: where f(x) � where g(y) � 0 0 0 0 en, from (6), it can be concluded that From (7), the adaptive law of parameters can be designed as follows:
As can be seen from Figures 3 and 4, if the control constant and gain constant are larger, the convergence speed of synchronization will be faster. e values of control constant and gain constant only affect the speed of synchronization. 1, 2, 2). (36) Remark 4. In [15], by adding dimensions to the drive system (26), the synchronization error form e i � y i − x i− 1 (i � 1, 2, 3, 4) of cross subtraction is constructed, and the appropriate controller and parameter adaptive laws are designed so that the drive system (26) and the response system (27) with unknown parameters can be synchronized in 4 D. By using eorem 1 and selecting Θ � ρ � 0, ρ′ � 0, the controller and parameter adaptive laws designed in this paper are the same as that in [15]; that is, the Θ − Φ synchronization in this paper can be transformed into the synchronization scheme in [15]. Compared with the conclusion given in [15], the constant gain matrices and gain function are added to the designed parameter adaptive laws, making eorem 1 and Corollary 1 given in this paper more universal.
Remark 5. In [23], the global fixed-time synchronization of four-dimensional hyperchaotic system (37) and two-dimensional chaotic Duffing system (38) is studied. By using eorem 1 and selecting Θ � , the synchronization type in this paper is the same as that in [23]. However, the authors in [23] focus on the problem of global fixed-time synchronization. When estimating unknown parameters, due to many constraints, the controller designed in [23] is more complex and difficult to meet in practice. Compared with the controller given in [23], the controller designed in this paper is relatively simple. At the same time, compared with the parameter adaptive laws given in [23], the constant gain matrices and the gain function are added in the designed parameter adaptive laws, which make the parameter estimation function given in this paper more effective.

Conclusions
Based on Lyapunov stability theory and adaptive control method, the synchronization of chaotic systems with different dimensions is researched in this paper. Using two scaling matrices, an adaptive feedback hybrid controller and parameter adaptive laws are designed to realize the synchronization of chaotic systems with different dimensions. According to the characteristics of unknown parameters, a more simplified synchronization control scheme is designed. Finally, three examples of chaotic systems are used to demonstrate the effectiveness of the proposed method. Although a new controller is set up according to the characteristics of the system, it is still more complex. It is hoped that a more simple and convenient controller is designed for the synchronization of chaotic system.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.