Anti-Synchronization Problem of the 4D Financial Hyper-Chaotic System with Periodically External Disturbance

: This paper is concerned with the partial anti-synchronization of the 4D ﬁnancial hyperchaotic system with periodically external disturbance. Firstly, the existence of the partial anti-synchronization problem for the nominal 4D ﬁnancial system is proven. Then, a suitable ﬁlter is presented, by which the periodically external disturbance is asymptotically estimated. Moreover, two disturbance estimator (DE)-based controllers are designed to realize the partial anti-synchronization problem of such a system. Finally, numerical simulation veriﬁes the effectiveness and correctness of the proposed results.


Introduction
Since Pecora and Carroll first presented chaos synchronization [1] in 1990, the synchronization problem of chaotic systems has been widely applied in engineering, science, and communications [2][3][4][5][6]. Many types of synchronization problems have been examined, such as complete synchronization, anti-synchronization, lag synchronization, projective synchronization, and so on. Anti-synchronization [7][8][9][10][11][12][13] is an important type of synchronization. It is a phenomenon whereby the state vectors of the slave systems have the same amplitude but opposite signs to those of the master system, i.e., the sum of two signals is expected to converge to zero when anti-synchronization appears. This type of synchronization has found distinctive applications in some fields, such as in communication systems, in which the system's security and secrecy can be deeply strengthened by transforming from complete synchronization and anti-synchronization periodically in the process of digital signal transmission. As is shown in [7], the existence of the anti-synchronization problem is f (−x) = − f (x) for a given chaotic systemẋ = f (x). It should be pointed out that the famous Lorenz system does not meet this necessary condition. A natural question arises as to whether the variables of chaotic systems meet such a condition or not. Recently, the partial anti-synchronization [14] was presented, which is a new kind of synchronization and is introduced in the next section.
Consider the following nonlinear systeṁ where x ∈ R n is the state, and f (x) ∈ R n is a continuous function.
There exists a non-singular transformation (X, Z) T = Px by which the system (1) can be divided into the following two subsystems: where P ∈ R n×n is a constant matrix, X ∈ R m , 1 ≤ m < n, Z ∈ R n−m , M(X, Z) ∈ R m meets M(−X, Z) = −M(X, Z), and G(X, Z) ∈ R n−m .
The controlled system (2) is presented as follows: where X and M(X, Z) are given in Equation (2), B ∈ R m×r , 1 ≤ r < n is a constant matrix, (M(X, Z), B) is assumed to be controllable according to Z, and U ∈ R r , r ≥ 1 is the controller to be designed.
Let the system (4) be the master system, and the slave system is given aṡ where Y ∈ R m and Z is given in Equation (3). Next, let E = X + Y, and the sum system is given aṡ where E ∈ R m is the state, and B, U are given in Equation (4).

Definition 1 ([14]
). For the system (6), if lim t→∞ E(t) = 0, then the master system (4) and the slave system (5) are considered to be anti-synchronized, which implies that the partial antisynchronization problem of the system (1) is realized.
It should be pointed out that among the abovementioned results, the external disturbance is not considered completely. Unfortunately, there are still many deficiencies in the current research, and in practice this is not the case. For the robust stabilization problems of the nonlinear systems, many methods have been reported, e.g., [15,16]. However, the obtained results are only robust, i.e., the disturbance is not asymptotically estimated by the observers or filters. More importantly, the robust control and disturbance rejection problems in the abovementioned works are mainly solved by the linear matrix inequalities (LMIs). It is well known that the stability conditions that are derived by the LMIs are only sufficient conditions and often result in conservative conclusions. Recently, the uncertainty and disturbance estimator (UDE)-based control method [17][18][19][20][21][22][23] emerged as an effective method to deal with the model uncertainty and external disturbance, and it has been widely applied in various systems. However, for the periodically external disturbance A sin(t) + C, where A and C are unknown constants, the proposed filter in [17] cannot asymptotically estimate such disturbance, so the obtained results are only robust, or practical. It is important to propose a suitable filter that can asymptotically estimate the periodically external disturbance. Inspired by the UDE-based control method, this paper shall propose a suitable filter in Section 3.
On the other hand, the novel chaotic finance system [24] was presented in 2001. This system model is described by three state variables of the time variations: x 1 stands for the interest rate; x 2 and x 3 are the investment demand and the price index, respectively. Recently, a four-dimensional finance system [25,26] was proposed. There are four subblocks that construct the system model, i.e., money, production, labor force, and stock. In our previous work [27], some control problems have been investigated by the adaptive control method. However, the impact of disturbance, especially the periodically external disturbance, is not considered. Therefore, it is very important and necessary to study the partial anti-synchronization problem of the 4D financial hyper-chaotic system with periodically external disturbance.
The main contributions of this paper are summarized as follows (1). The existence of the partial anti-synchronization problem of the 4D financial system is proven; (2). A suitable filter that can asymptotically estimate the periodically external disturbance is proposed; (3). Two DE-based controllers are designed and used to realize the partial anti-synchronization problem.
The rest of this paper is organized as follows. Section 2 introduces the problem formation, Section 3 presents the main results of this paper, Section 4 provides the numerical simulation results, and Section 5 gives the conclusions.
For subsequent use, the dynamic feedback control method is introduced in the next section.

Lemma 1 ([28]
). Consider the following systeṁ where q ∈ R n is the state; b ∈ R n×l , l ≥ 1, and u is the controller to be designed. If (H(q), b) can be stabilized, then the controller u is

Problem Formation
The controlled 4D hyper-chaotic system with periodically external disturbance is given asẋ u ∈ R 1 is the controller to be designed, w 1 (t) = A sin(t) + C, and A, C are unknown constants.
The main goal of this paper is to investigate the partial anti-synchronization problem of the system (10), and present some new results.

The Existence of the Partial Anti-Synchronization Problem
In this section, the existence of the partial anti-synchronization problem for the system (12) is investigated and a conclusion is derived as follows. Theorem 1. Considering the system (12), there exists a non-singular transformation: by which the system (12) can be divided into the following two subsystemṡ where X ∈ R 3 , Z ∈ R 1 , and which implies that the partial anti-synchronization problem of the system (12) exists.
The proof of Theorem 1, please see Appendix A.1. By the transformation (13), the system (10) is also divided into the following subsystemṡ and the subsystem given in Equation (15), where and U ∈ R 1 is the controller to be designed. Let the subsystem (18) be the master system, and the slave system is presented as followṡ where Y ∈ R 3 and Z ∈ R 1 is given in Equation (15), Let E = X + Y, and the sum system is given aṡ where E ∈ R 3 is the state, and B, U are given in Equation (18). Next, the controller U = U s + U w is designed by two steps. In the first step, U s is proposed for the nominal systemĖ = M(Z)E + BU s , and U w is obtained by designing a suitable filter G f (s) for the periodically external disturbance in the second step.

The Controllers Are Designed for the Nominal System
Theorem 2. Regarding the nominal systemĖ = M(Z)E + BU s , the dynamic feedback controller U s is proposed as follows U s = K(t)E where K(t) = k(t)B T , and The proof of Theorem 2, please see Appendix A.2. It is noted that the nominal systemĖ = M(Z)E + BU s has a simple form; another linear feedback controller is proposed in the next step.
The proof of Theorem 3, please see Appendix A.3.
The proof of Theorem 4, please see Appendix A.4.

The DE-Based Controller Is Designed for Periodically External Disturbance
Theorem 5. For the sum system (21), the DE-based controller U w is presented as follows The proof of Theorem 5, please see Appendix A.5.

Comparison with the Effect of Parameters of Periodically External Disturbance
The initial conditions of the master subsystem given in Equation (18) and the slave subsystem given in Equation (20) are chosen as X(0) = [1, 2, 3], Y(0) = [−2, −1, −2], respectively, and Z(0) = 1. Without loss of generality, the controller U s given in Equation (24) is taken as an example.
From Figure 1, it can be seen that the sum system is asymptotically stable. Figure 2 shows the states of the master subsystem and the slave subsystem, respectively. It is easy to see that the state vectors of the slave subsystem have the same amplitude but opposite signs to those of the master subsystem, which implies that the partial anti-synchronization problem of the 4D financial hyper-chaotic system is realized. Figure 3 shows thatŴ 1 (t) tends to W 1 (t) as t → +∞. Figure 4 shows the state of control signal U(t).  Case 2: A = 100, C = 20. Figure 5 shows that the sum system is asymptotically stable. Figure 6 shows the states of the master subsystem and the slave subsystem, respectively. It is easy to see that the state vectors of the slave subsystem have the same amplitude but opposite signs to those of the master subsystem, which implies that the partial anti-synchronization problem of the 4D financial hyper-chaotic system is realized. Figure 7 shows thatŴ 1 (t) tends to W 1 (t) as t → +∞. Figure 8 shows the state of control signal U(t).

Remark 1.
From the above figures, it can be seen that the effect of the disturbance parameters (A and C) on the anti-synchronization problem is very small.

Comparison with the Same Control Strategy
The initial conditions of the master subsystem given in Equation (18) and the slave subsystem given in Equation (20) are chosen as X(0) = [1,2,3], Y(0) = [−2, −1, −2], respectively, and Z(0) = 1, and A = 100, C = 20. The controller U s given in Equation (24) is chosen here. Figure 9 shows that the sum system is asymptotically stable. Figure 10 shows the states of the master subsystem and the slave subsystem, respectively. It is easy to see that the state vectors of the slave subsystem have the same amplitude but opposite signs to those of the master subsystem, which implies that the partial anti-synchronization problem of the 4D financial hyper-chaotic system is realized. Figure 11 shows thatŴ 1 (t) tends to W 1 (t) as t → +∞. Figure 12 shows the state of control signal U(t).

Remark 2.
From the above figures, it can be seen that the effect of the controller U s given in Equation (24) is stronger than that given in Equation (22).

Conclusions
The partial anti-synchronization problem of the 4D financial hyper-chaotic system with periodically external disturbance has been investigated. Firstly, the partial antisynchronization problem of the nominal 4D financial system has been proven. Then, by designing a suitable filter, two disturbance estimator (DE)-based controllers have been presented and used to realize the partial anti-synchronization problem. It has been proven that the proposed filter can asymptotically estimate the periodically external disturbance. Finally, the effectiveness and correctness of the proposed results have been verified by numerical simulation. Proof. Let β = Diag{β 1 , β 2 , · · · , β 4 }, |β i | = 0, 1, i = 1, 2, · · · , 4, and it is easy to judge that is a solution of the following equations about β i.e.,    β 1 β 2 = β 1 and β 1 = β 3 and β 1 = β 4 β 1 = β 3 β 1 β 2 = β 4 (A3) Thus, the transformation (13) is obtained, which completes the proof. Proof. Substituting the controller U s given in Equation (24) into the nominal systemĖ = M(Z)E + BU s , we find thaṫ is asymptotically stable, which completes the proof.