# Projective Synchronization of Hyperchaotic Financial Systems

Authors: Jiming Zheng; Bingfang Du
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

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.


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## Body

## 1. 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, drive-response 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 integer-order 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.


## 2. Projective Synchronization of Hyperchaotic Financial Systems

Consider the financial system [7] as follows:
(1)x˙=z+y-ax,y˙=1-by-x2,z˙=-x-cz,
where x, y, and z represent the interest rate, investment demand, and price index, respectively. The parameter a is the saving, b is the per-investment cost, and c 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 w. Therefore, the following improved chaotic finance system is constructed [8]:
(2)x˙=z+y-ax+w,y˙=1-by-x2,z˙=-x-cz,w˙=-dxy-kw,
where a, b, c, d, and k are the parameters of the system, and they are positive numbers. When a=0.9, b=0.2, c=1.5, d=0.2, and k=0.17, system (2) presents hyperchaotic behavior [8]. When the initial value of system (2) is taken as (1,3,2,5) and t∈(0,1000), the phase portraits of system (2) are shown in Figures 1(a)–1(d).Phase portraits of hyperchaotic finance system (2).
(a)
3D view in thex-y-z space
(b)
3D view in thex-y-w space
(c)
Projection onto thex-y plane
(d)
Projection onto thez-w planeChaos 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:
(3)x˙y˙z˙w˙=-a0110-b00-10-c0000-kxyzw+0100+xy-x20-dxy.In order to facilitate design of controller, using the thought of state transition, we construct the following response system for the given system (3):
(4)x˙sy˙sz˙sw˙s=-a0110-b00-10-c0000-kxsyszsws+α0100+xy-x20-dxy+u,
where α is a scaling factor and u is the external input control vector.Denotex=(x,y,z,w)T, y=(xs,ys,zs,ws)T, and e=(e1,e2,e3,e4)T. If we define the error system as e=y-αx, we can get the error system
(5)e˙=Ae+u,
where
(6)A=-a0110-b00-10-c0000-k.Theorem 1.
If we design the controlleru=Be for the error system (5), then the system (5) is asymptotically stable at the origin, where (7)B=k1·b11k1k1·-a1k2k1·1k3-1k1-a2k3k1·-1k1-a3k4k2·a1k2k2·b22k2k2·-a4k3k2·-a5k4k3·a2k3k3·a4k3k3·b33k3k3·-a6k4k4·a3k4k4·a5k4k4·a6k4k4·b44k4,b11≤a,b22≤b,b33≤c,b44≤k;ai∈Ri=1,2,3,4,5,6,ki∈R+i=1,2,3,4.In order to prove Theorem1, we introduce Lemma 2 firstly.Lemma 2.
Suppose a dynamic system can be written as(8)x˙1x˙2⋮x˙n=k1a11k1a12⋯k1a1nk2a21k2a22⋯k2a2n⋮⋮⋮⋮knan1knan2⋯knannx1x2⋮xn.
If the system (8) satisfies the following conditions [9]: (1)
∀aij∈R,(2)
aij=-aji(i≠j),(3)
aii≤0 (not all aii are equal to zero),(4)
∀ki>0,
then the states of system (8) will decrease to zero gradually.Prove Theorem1 as follows.Proof.
Let controller beu=Be, where B=(bij)4×4 is a 4×4 order constant matrix to be designed; e=(e1,e2,e3,e4)T. We rewrite the error system (5) as
(9)e˙=c+Be,
where(10)C=cij4×4=k1·-ak10k1·1k1k1·1k10k2·-bk200k3·-1k30k3·-ck30000k4·-kk4.
In order to make the system (9) satisfy Lemma 2, design matrix B to satisfy conditions (11) and (12) as follows:
(11)-ak1+b11k1≤0,-bk2+b22k2≤0,-ck3+b33k3≤0,-kk4+b44k4≤0,(12)b12k1=-b21k2,1k1+b13k1=--1k3+b31k3,1k1+b14k1=-b14k4,b23k2=-b32k3,b24k2=-b42k4,b34k3=-b43k4.
It can be seen from (11) that b11≤a, b22≤b, b33≤c, and b44≤k.
Let the coefficient matrix of (12) be H, and let its augmented matrix be H-. After calculation, we can get r(H)=r(H-), and the numbers of unknowns are greater than the numbers of (12), so (12) has infinitely many solutions. Note that all solutions of (12) are as follows:
(13)η=b12,b21,b13,b31,b14,b41,b23,b32,b24,b42,b34,b43=-a1k1k2,a1,k1k3-1-a2k1k3,a2,-1-a3k1k4,a3,-a4k2k3,a4,-a5k2k4,a5,-a6k3k4,a6,
where ai∈Ri=1,2,3,4,5,6; ki∈R+i=1,2,3,4.
Therefore integrate (11) and (12) to obtain the matrix B: (14)B=k1·b11k1k1·-a1k2k1·1k3-1k1-a2k3k1·-1k1-a3k4k2·a1k2k2·b22k2k2·-a4k3k2·-a5k4k3·a2k3k3·a4k3k3·b33k3k3·-a6k4k4·a3k4k4·a5k4k4·a6k4k4·b44k4.So system (9) becomes(15)e˙1e˙2e˙3e˙4=k1·-ak1+b11k1k1·-a1k2k1·1k3-a2k3k1·-a3k4k2·a1k2k2·-bk2+b22k2k2·-a4k3k2·-a5k4k3·-1k3+a2k3k3·a4k3k3·-ck3+b33k3k3·-a6k4k4·a3k4k4·a5k4k4·a6k4k4·-kk4+b44k4e1e2e3e4.According to Lemma 2, the system (15) is asymptotically stable at the origin; namely, the system (5) is asymptotically stable at the origin.Remark 3.
The error system (5) is asymptotically stable at the origin; namely, the system (3) and system (4) achieve the projective synchronization.
We takeb11=0.5, b22=0.1, b33=1, and b44=0.1; a1=1, a2=2, a3=3, a4=4, a5=5, and a6=6; k1=1, k2=2, k3=3, and k4=4. Then if we take a=0.9, b=0.2, c=1.5, d=0.2, k=0.17, and α=1/2; the initial values of drive system (3) and response system (4) are taken as (1,2,0.5,0.5) and (1,3,2,5), respectively. So the initial value of error system (5) is (0,1,1.5,4.5), and the state variables of system (5) varying with time t are shown in Figures 2(a)–2(e).Time evolutions of the state variables of system (5).
(a)
Time evolutions ofxs and x
(b)
Time evolutions ofys and y
(c)
Time evolutions ofzs and z
(d)
Time evolutions ofws and w
(e)
Time evolutions ofe1,e2,e3, and e4Remark 4.
To make the structure of the controller as simple as possible, we takebii=0, i=1,2,3,4, ai=0, i=1,2,…,6, and ki=1, i=1,2,3,4. So we can get control matrix
(16)B=00k1·1k3-1k1k1·-1k1000000000000,ki∈R+,i=1,2,3,4
which is a special case.Remark 5.
Whenα=1, the system (3) and system (4) are of complete synchronization [10]; when α=-1, the system (3) and system (4) are antisynchronization [11].Remark 6.
Whenk1=k2=k3=k4=1, the coefficient matrix of system (15) is the antisymmetric matrix in [12]. That is to say, they have the same control scheme.


## 3. 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 wt and constructed a new hyperchaotic financial system:
(17)x˙t=-a′xt+yt+wt,yt′=-yt-a′xtzt,z˙t=b′+a′xtyt,w˙t=-c′xtzt-d′wt,
where a′ and b′ are parameters of the system (17), c′ is a constant (c′=0.2), and d′ is a control parameter. And when a′=3, b′=15, c′=0.2, and d′=0.12, system (17) presents hyperchaotic behavior [13]; the initial value of system (17) is taken as (1.5,3.2,2,3); take t∈(0,1000); the phase portraits of system (17) are shown in Figures 3(a)–3(d).Phase portraits of hyperchaotic finance system (17).
(a)
3D view in thex-y-z space
(b)
3D view in thex-y-w space
(c)
Projection onto thex-y plane
(d)
Projection onto thez-w planeLet the system (3) be the drive system, and let the system (17) be response system. So the system (17) becomes
(18)x˙ty˙tz˙tw˙t=-a′-a′010-1000000000-d′xtytztwt+0-a′xtzta′xtyt-c′xtzt+00b′0+u.
We denote
(19)A=-a0110-b00-10-c0000-k,h1=0100,fx=xy-x20-dxy,x=xyzw,D=-a′-a′010-1000000000-d′,h2=00b′0,gy=0-a′xtzta′xtyt-c′xtzt,y=xtytztwt.Letu=(A-D)y+αh1+f(x)-h2-g(y)+Ee, where E=(eij)4×4 is a 4×4 order constant matrix to be designed, and e=e1,e2,e3,e4Τ. So the response system (18) becomes
(20)x˙ty˙tz˙tw˙t=-a0110-b00-10-c0000-kxtytztwt+α0100+xy-x20-dxy+Ee.We also define the error system ase=y-αx. Let the system (20) minus the α times of the system (3); we get the error system
(21)e˙1e˙2e˙3e˙4=-a0110-b00-10-c0000-ke1e2e3e4+Ee1e2e3e4.Theorem 7.
For the error system (21), let the controller be
(22)u=A-Dy+αh1+fx-h2-gy+Ee,
where(23)E=k1·e11k1k1·-a1k2k1·1k3-1k1-a2k3k1·-1k1-a3k4k2·a1k2k2·e22k2k2·-a4k3k2·-a5k4k3·a2k3k3·a4k3k3·e33k3k3·-a6k4k4·a3k4k4·a5k4k4·a6k4k4·e44k4,e11≤a,e22≤b,e33≤c,e44≤k,ai∈R,i=1,2,…,6;ki∈R+,i=1,2,3,4.Then the system (21) is asymptotically stable at the origin.Proof.
Ifu=(A-D)y+αh1+f(x)-h2-g(y)+Ee, according to the definition of the error system, the error system of the driven system (3) and response system (18) is system (21); namely,
(24)e˙=C+Ee.
System (24) has the same structure with the system (9), so we can get E=B; namely,(25)E=k1·(e11k1)k1·-a1k2k1·1k3-1k1-a2k3k1·-1k1-a3k4k2·a1k2k2·e22k2k2·-a4k3k2·-a5k4k3·a2k3k3·a4k3k3·e33k3k3·-a6k4k4·a3k4k4·a5k4k4·a6k4k4·e44k4.Therefore, the system (24) becomes
(26)e˙=C+Ee,
where(27)C+E=k1·-ak1+e11k1k1·-a1k2k1·(1k3-a2k3)k1·-a3k4k2·a1k2k2·(-bk2+e22k2)k2·-a4k3k2·-a5k4k3·-1k3+a2k3k3·a4k3k3·-ck3+e33k3k3·-a6k4k4·a3k4k4·a5k4k4·a6k4k4·-kk4+e44k4.According to Lemma 2, the system (26) tends to zero gradually; namely, the system (21) is driven to the origin gradually.Remark 8.
The system (21) tended to zero gradually; that is, the drive system (3) and response system (18) achieve the projective synchronization.
Generally, we takee11=0.5, e22=0.1, e33=1, and e44=0.1; a1=1, a2=2, a3=3, a4=4, a5=5, and a6=6; k1=1, k2=2, k3=3, and k4=4. And we take a=0.9, b=0.2, c=1.5, d=0.2, k=0.17, and α=-1; the initial values of drive system (3) and response system (17) are taken as 0.1,-0.1,0.1,0.1 and 0,1,0.2,0.5, respectively. So the initial value of error system (21) is -0.1,1.1,0.1,0.4, and the state variables of system (21) varying with time t are shown in Figures 4(a)–4(e).Time evolutions of the state variables of system (3) and system (20).
(a)
Time evolutions ofxt and x
(b)
Time evolutions ofyt and y
(c)
Time evolutions ofzt and z
(d)
Time evolutions ofwt and w
(e)
Time evolutions ofe1,e2,e3, and e4


## 4. 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.



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