Stabilization and synchronization of hyper-chaotic financial system involved in positive interest rate

In real financial market, the delayed market feedback and the delayed effect of government macro-control are inevitable. And both the delay of market feedback and the delay of macro-control effect bring about the mathematical difficulties in studying stabilization and synchronization of the hyper-chaotic financial system. However, employing Lyapunov function method, differential mean value theorem, suitable bounded hypotheses and pulse control technology results in the globally asymptotical stabilization and synchronization criteria. It is the first paper to drive the stabilization and synchronization criteria under the assumptions of the double delays. Finally, numerical examples illuminate the effectiveness of the proposed methods.


Introduction
The complexity of economic systems often leads to unpredictable dynamic behavior. The periodic economic crisis reminds me that it is necessary to study and control the instability and chaos of financial system. Recently, many scholars have investigated the stability and synchronization of a class of chaotic finical system that is composed of the production sub-block, currency sub-block, securities sub-block, and labor sub-block (see, e.g. [1][2][3][4][5][6][7][8][9]). In many related literature( [1][2][3][4][5][6][7][8][9]), the following finical system was investigated, which is composed of the production sub-block, currency sub-block, securities sub-block, and labor sub-block, (1.1) where x represents the interest rate, y represents the investment demand, z represents the price index, a represents savings, b represents the unit investment cost, and c represents the elasticity of commodity demand.
Based on the improved financial chaos system model, the authors of [10] took the global economic crisis caused by American subprime mortgage crisis in 2007 as the background of the

System descriptions and preparations
From [10], there exist three equilibrium points for the system (1.2): Since the delayed feedback is a common phenomenon in the real market,I get the following delayed feedback model for the system (1.2), cd−ck ) with the positive interest rate θ > 0 of the system (1.2) corresponds to the zero solution of the following system

4)
where X = (x 1 , x 2 , x 3 , x 4 ) T . Furthermore, the system (2.4) can be rewritten as the following system in form of vector-matrix, Under the delayed impulse control on the system (2.5), I can get where X is defined in (2.3). That is, the zero solution of the system of (2.7) corresponds to the equilibrium point cd−ck ) of the hyper-chaotic financial system (2.8). Construct the following response system for the drive system (2.7), (2.9) and the error system of pulse synchronization is given as follows, Theorem 3.2]). Under the assumptions of Theorem 3.1, the following fuzzy system (2.17) is bounded under the meaning of L ∞ :

(2.12)
For convenience, I employ the following notations: • For a symmetric matrix A, I denote by λ max A the maximum eigenvalue of A ; • For a matrix A, I denote by A = λ max (A T A) the norm of A .

Stabilization by impulse control
At first, I may assume in this section that X(t − k ) = X(t k ) for all k ∈ Z + . Secondly, from Lemma 2.1, I can similarly derive the boundedness of the system (2.7) and (2.8) under mild conditions. So in this paper, I may give the following boundedness assumptions: If the upper limit of delays τ is appropriately small, the following continuity hypothesis is natural due to the boundedness.
(H2) For any given τ > 0, there exists the corresponding positive number c τ > 0 such that In this section, I assume that time delays τ(t) ∈ [−τ, 0], and ρ k ∈ [−ρ, 0] for all k ∈ Z + . In order to obtain the stability of the system, a certain pulse frequency is required, so I assume a smaller pulse interval as follows, sup where c 0 is a positive number. Mainly inspired by some methods and ideas of my another [20], I present the following Theorem.

4)
where d 0 is a positive scalar,

5)
and I represents the identity matrix, ρ k < t k − t k−1 , for all k ∈ Z + , then the zero solution of the system (2.7) is globally asymptotically stable, and the equilibrium solution P 1 with the positive interest rate θ of the system (2.8) is globally asymptotically stable.

Remark 2.
The condition ρ k < t k − t k−1 implies that every macro-control measure (pulse) of the government should be effective enough to see the pulse effect within each pulse interval. Besides, the condition sup k∈Z + (t k − t k−1 ) < c 0 guarantees pulse (Macro-control) of a certain frequency if c 0 > 0 is appropriate small. No matter how complex and chaos the financial system is, high-frequency active macro-control is conducive to the global asymptotical stability of the economic system. To derive the synchronization criterion, I may consider the following boundedness assumptions: (H3) There are two positive scalars N 1 , N 2 such that (H4) For any given τ > 0, there exists the corresponding positive number d τ > 0 such that Theorem 3.2. Assume that X, Y satisfy the boundedness conditions (H1) and (H2), and the error variable e satisfies the boundedness assumptions (H3)-(H4). If, in addition, the following condition holds, 10) then the system (2.9) can be globally exponentially synchronized onto the system (2.7), where Proof. Consider the following Lyapunov function (3.12) Similarly, I can conclude from the assumptions of Theorem 3.2 that

Remark 3.
It is the first paper to derive the synchronization criterion for the systems (2.9) and (2.7) under the assumptions of the double delays.

Numerical examples
Example 4.1. Consider the system (2.7) or (2.8) with the following data: which implies that the condition (3.4) is satisfied, and the equilibrium solution P 1 with the positive interest rate θ = 57.35% of the system (2.8) is globally asymptotically stable due to Theorem 3.1.  Table 1 illuminates that when the system reach stable, the higher the interest rate, the smaller the pulse interval. This shows, in order to reach a balance of higher interest rates in the financial market, the government should speed up the pace of macro-control of the economy. which implies that the condition (3.10) holds. And hence, Theorem 3.2 tells me that the system (2.9) can be globally exponentially synchronized onto the system (2.7).

Conclusions
Interest rates are always positive at most of countries in the world when the economies are in balance. So my Theorem 3.1 and Theorem 3.2 illustrate theoretical guidance significance for the actual financial market. In particular, Theorem 3.1 shows that positive and correct macroeconomic control measures with a certain frequency are conducive to market balance and high positive interest rates. Finally, two numerical examples shows the effectiveness and feasibility of stabilization and synchronization criteria.

Conflicts of Interest:
The author declares no conflict of interest.