Projective Synchronization of Hyperchaotic Financial Systems

. Based on a special matrix structure, the projective synchronization control laws of the hyperchaotic financial systems are proposed in this paper. Put a hyperchaotic financial system as the drive system, via transformation of the system state variables, construct its response system, and then design the controller based on the special matrix structure. The given scheme is applied to achieve projective synchronization of the different hyperchaotic financial systems. Numerical experiments demonstrate the effectiveness of the method.


Introduction
Chaos in the economy was first discovered since 1985; it had made a tremendous impact on market economy.Chaos theory provides new approaches and ideas for financial crisis and other related issues.Synchronous development of the financial system is a real problem faced by many economists, such as how to achieve synchronous development in different countries and areas.In 1999, Mainieri and Rehacek [1] proposed a chaotic synchronous mapping; namely, driveresponse system can be synchronous with a desired scaling factor.In recent decades, projective synchronization [2,3] of the chaotic financial system is also of concern.Scholars realized projective synchronization of three-dimensional fractional-order chaotic financial system and the integerorder one, respectively.Hyperchaotic system has two or more positive Lyapunov exponents.The economic system is a high-dimensional nonlinear system, whose chaos is mostly super chaos [4].Achieving synchronization of hyperchaotic financial system is closer to the actual condition, so some researchers have investigated the synchronization of hyperchaotic financial system [5,6].In this paper, based on a special matrix and conversion of state variables, we will realize projective synchronization of the identical and different hyperchaotic financial systems.

Projective Synchronization of Hyperchaotic Financial Systems
Consider the financial system [7] as follows: where , , and  represent the interest rate, investment demand, and price index, respectively.The parameter  is the saving,  is the per-investment cost, and  is the elasticity of demands of commercials.And they are positive constants.
Based on the chaotic finance of system (1), scholars found that the factors affecting the interest rate are related not only to investment demand and price index, but also to the average profit margin denoted by .Therefore, the following improved chaotic finance system is constructed [8]: where , , , , and  are the parameters of the system, and they are positive numbers.When  = 0.9,  = 0.2,  = 1.5,  = 0.2, and  = 0.17, system (2) presents hyperchaotic behavior [8].When the initial value of system (2) is taken as (1, 3, 2, 5) and  ∈ (0, 1000), the phase portraits of system (2) are shown in Figures 1(a Chaos of financial systems is mostly hyperchaos.In order to achieve financial synchronous development of developed and developing countries or different areas, we need to solve more problems of synchronization of hyperchaotic financial systems.
The projective synchronization discussed in this paper is defined as two relative chaotic dynamical systems can be synchronous with a desired scaling factor [2].
To discuss projective synchronization of the hyperchaotic financial system (2), we rewrite the system (2) as follows: In order to facilitate design of controller, using the thought of state transition, we construct the following response system for the given system (3): where  is a scaling factor and u is the external input control vector.Denote x = (, , , ) T , y = (  ,   ,   ,   ) T , and e = ( 1 ,  2 ,  3 ,  4 ) T .If we define the error system as e = y − x, we can get the error system where Theorem 1.If we design the controller u = e for the error system (5), then the system (5) is asymptotically stable at the origin, where In order to prove Theorem 1, we introduce Lemma 2 firstly.
Lemma 2. Suppose a dynamic system can be written as

𝑥 𝑛
) .(8) If the system (8) satisfies the following conditions [9]: ≤ 0 (not all   are equal to zero), then the states of system ( 8) will decrease to zero gradually.Prove Theorem 1 as follows.
According to Lemma 2, the system (15) is asymptotically stable at the origin; namely, the system (5) is asymptotically stable at the origin.

Projective Synchronization of Two Different Hyperchaotic Financial Systems
In the real world, we will not only meet synchronization of two identical hyperchaotic financial systems, but also encounter more often the one of two different hyperchaotic financial systems.Here we consider the projective synchronization of the system (2) and the following system (17).Ding et al. [13] introduced a state feedback controller   and constructed a new hyperchaotic financial system: where   and   are parameters of the system (17),   is a constant (  = 0.2), and   is a control parameter.And when   = 3,   = 15,   = 0.2, and   = 0.12, system (17) presents hyperchaotic behavior [13]; the initial value of system (17) is taken as (1.5, 3.2, 2, 3); take  ∈ (0, 1000); the phase portraits of system (17) are shown in Figures 3(a)-3(d).
Proof.If u = ( − )y + (h 1 + (x)) − h 2 − (y) + e, according to the definition of the error system, the error system of the driven system (3) and response system (18) is system (21); namely, System (24) has the same structure with the system (9), so we can get  = ; namely, Therefore, the system (24) becomes where ) . ( According to Lemma 2, the system (26) tends to zero gradually; namely, the system (21) is driven to the origin gradually.

Conclusions
Based on a special matrix structure and the conversion of the state variable, we design the controllers to achieve the projective synchronization of hyperchaotic financial systems in this paper.And the control law is also suitable for synchronous control of other chaotic systems.Theoretical analysis and simulation results show the effectiveness of the method.