# Qualitative Study of a 4D Chaos Financial System

Authors: Fuchen Zhang; Gaoxiang Yang; Yong Zhang; Xiaofeng Liao; Guangyun Zhang
Journal: Complexity
 (2018)

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## Abstract

Some dynamics of a new 4D chaotic system describing the dynamical behavior of the finance are considered. Ultimate boundedness and global attraction domain are obtained according to Lyapunov stability theory. These results are useful in estimating the Lyapunov dimension of attractors, Hausdorff dimension of attractors, chaos control, and chaos synchronization. We will also present some simulation results. Furthermore, the volumes of the ultimate bound set and the global exponential attractive set are obtained.


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## Body

## 1. Introduction

Chaotic systems are characterized by their extreme sensitivity both to initial conditions and to system parameters. The study of various aspects of chaotic systems has received great interest among scientists from various fields due to their numerous potential applications in science and engineering [1–30]. Among such aspects, the problem of investigating the ultimate boundedness of chaotic systems and hyperchaotic systems is an important subject. Since the discovery of the famous Lorenz chaotic attractor [1], ultimate boundedness of the Lorenz attractor has been investigated by Leonov et al. in a series of articles [7, 10]. Since then many papers have studied ultimate boundedness of other chaotic systems [25–32]. However, the approach taken in each is only suitable for that particular chaotic system. It is very difficult to propose a universal approach to estimate the bounds for an arbitrary chaotic system. Zhang et al. studied the ultimate boundedness of the Lü system and the Chen system by using the Lyapunov stability theory and optimization method [25–27]. Particularly, Liao et al. studied the ultimate boundedness of the Yang-Chen system by using geometric and algebraic methods [31]. Zhang et al. studied the ultimate boundedness of a novel finance chaotic system by using the Lyapunov stability theory and iterative method [32]. To this end, it is necessary to study the bounds of the new chaotic systems.Recently, a financial chaos dynamical system is reported as follows [32, 33]:
(1)dxdt=z+y−ax,dydt=1−by−x2,dzdt=−x−cz.System (1) models a financial dynamical system composed of product, money, bond, and labor force. The variables x, y, and z denote the interest rate, the investment demand, and the price index, respectively. The positive parameters a, b, and c denote the saving amount, the per investment cost, and the demand elasticity of commercials, respectively. The factors that induce the change of the interest rate x mainly come from two aspects: the price index and the surplus between the investment demand and the savings amount. The changing rate of y is determined by the benefit rate of investment (we assume that the rate is constant during a certain period), the feedback of the investment demand, and the interest rate. The change of the price index z is controlled by the real interest rate and price index.According to the financial dynamical system (1), by adding a variable u, one gets a new 4D financial chaos dynamical system as follows:
(2)dxdt=z+y−ax,dydt=1−by−x2,dzdt=−x−cz+du,dudt=−ku−mz.where the variable u denotes control input and economically state intervention to balance the economic environment and a,b,c,d,k,m are positive real parameters of system (2).The Lyapunov exponents of the financial system (2) are calculated numerically for the parameter values a=0.9, b=0.2, c=1.5, d=1, k=0.05, and m=0.005 with the initial state x0,y0,z0,u0=0,1,…,0.5,0. System (2) has the Lyapunov exponents as λLE1=0.1740, λLE2=0.1314, λLE3=0.0000, and λLE4=−15.6059 for the above parameters (see [8, 9] for a detailed discussion of Lyapunov exponents of strange attractors in chaos dynamical systems).The chaotic attractor of system (2) in xOyz space for the positive parameters a=0.9, b=0.2, c=1.5, d=1, k=0.05, and m=0.005 is shown in Figure 1.Remark 1.
The chaotic attractor is one of the important concepts of dynamical systems. Although system (2) has a chaotic attractor for the positive parameters a=0.9, b=0.2, c=1.5, d=1, k=0.05, and m=0.005, the type of the chaotic attractor of system (2) is still unknown. It is interesting to discuss whether the chaotic attractor of system (2) is hidden or self-excited in the future (see the excellent papers [17, 19] for a detailed discussion of the chaotic attractor of dynamical systems).
This paper is organized as follows. In the next section, we will study the bounds for the chaotic attractors (2) using the Lyapunov stability theory and optimization method. To validate the ultimate bound estimation, numerical simulations are also provided. Finally, we will give some concluding remarks in Section 3.Figure 1
The chaotic attractor of system (2) in xOyz space with a=0.9, b=0.2, c=1.5, d=1, k=0.05, m=0.005, and x0,y0,z0,u0=0,1,…,0.5,0.


## 2. Main Properties

Theorem 1.
Suppose thata>0,b>0,c>0, k>0,d≠0,m>0.
LetXt=xt,yt,zt,ut be an arbitrary solution of system (2). Then, the following set
(3)Ω=x,y,z,umx2+my2+mz2+du2≤R2,is the ultimate bound set and positively invariant set of system (2), where
(4)R2=m4ab−a,c≥a,k≥a,b≥2a,m4cb−c,a>c,k>c,b≥2c,m4kb−k,a>k,c>k,b≥2k,mb2,b<2a,b<2c,b<2k.Proof 1.
Define the Lyapunov-like function(5)VX=mx2+my2+mz2+du2,where Xt=xt,yt,zt,ut. Computing the derivative of VX along the trajectory of (2), we have
(6)dVXdt2=2mxdxdt+2mydydt+2mzdzdt+2dududt=2mxz+xy−ax+2my1−by−x2+2mz−x−cz+du+2du−ku−mz=−2amx2−2bmy2+2my−2cmz2−2dku2=−2amx2−2bmy−12b2−2cmz2−2dku2+m2b.
Obviously, the surfaceΓ1, that is, defined by
(7)Γ1=Xamx2+bmy−12b2+cmz2+dku2=m4b,is an ellipsoid in R4 ∀a>0, b>0, c>0, k>0, d≠0, and m>0. Outside Γ1:dVX/dt<0, while inside Γ1:dVX/dt>0. Thus, the maximum value of VX can only be reached on Γ1. Since the VX is a continuous function and Γ1 is a bounded closed set, then the function (5) can reach its maximum value maxX∈Γ1VX=R2 on the surface Γ1 defined in (7).
Obviously,X∣VX≤maxX∈Γ1VX=R2 contains the solutions of system (2). By solving the following conditional extremum problem, one can get the maximum value of the function (5) as follows:
(8)maxVX=maxmx2+my2+mz2+du2,s.t.amx2+bmy−12b2+cmz2+dku2=m4b.
That is to say,(9)maxVX=maxmx2+my2+mz2+du2,s.t.mx2m/4ba+my−1/2b2m/4b2+mz2m/4bc+du2m/4bk=1.
Let us takemx=x1,my=y1,mz=z1,du=u1 as the new variables, then (9) transforms into the following form:
(10)maxVX=maxx12+y12+z12+u12,s.t.x12m/4ba+y1−m/2b2m/4b2+z12m/4bc+u12m/4bk=1.
According to the Lagrange multiplier method, we can easily get the above conditional extremum problem (10) as
(11)R2=m4ab−a,c≥a,k≥a,b≥2a,m4cb−c,a>c,k>c,b≥2c,m4kb−k,a>k,c>k,b≥2k,mb2,b<2a,b<2c,b<2k.
Finally, it is easy to show that (3) is the ultimate bound set and positively invariant set of system (2).This completes the proof.Remark 2.
(i)
Whena=0.9,b=0.2,c=1.5,d=1,k=0.05, and m=0.005, then we can obtain that
(12)Ω1=x,y,z,u0.005x2+0.005y2+0.005z2+u2≤0.412,which is the ultimate bound set and positively invariant set of system (2) according to Theorem 1. Figure 2 shows the chaotic attractor of system (2) in the xOyz space defined by Ω1.
(ii)
We can figure out that the volume of the setΩ in (3) is vΩ=π2R4/2mmd according to Theorem 1.Figure 2
The chaotic attractor of system (2) in the xOyz space defined by Ω1 with a=0.9,b=0.2,c=1.5,d=1,k=0.05, and m=0.005.Though Theorem1 gives the ultimate bound set and positively invariant set of the financial chaos system (2), it does not give the global exponential attractive set of system (2). The global exponential attractive set of system (2) is described by the following Theorem 2.Theorem 2.
Suppose thata>0,b>0,c>0,k>0,d≠0, and m>0, and let
(13)VX=mx2+my2+mz2+du2,L2=mθb,θ=mina,b,c,k>0.
Then, the estimation,(14)VXt−L2≤VXt0−L2e−θt−t0,holds for system (2), and thus
(15)Ψ=XVX≤L2,which is the globally exponential attractive set of system (2), that is, lim¯t→+∞VXt≤L2.Proof 2.
Define the following function of a variable(16)fy=−bmy2+2my,then we can get
(17)maxy∈Rfy=mb.
Define the Lyapunov function(18)VX=mx2+my2+mz2+du2.
Differentiating the Lyapunov functionVX in (18) with respect to time t along the trajectory of system (2), when VXt>L2 and VXt0>L2, we have
(19)dVXdt2=2mxdxdt+2mydydt+2mzdzdt+2dududt=2mxz+xy−ax+2my1−by−x2+2mz−x−cz+du+2du−ku−mz=−2amx2−2bmy2+2my−2cmz2−2dku2≤−amx2−bmy2−cmz2−dku2−bmy2+2my≤−amx2−bmy2−cmz2−dku2+fy≤−amx2−bmy2−cmz2−dku2+maxy∈Rfy≤−amx2−bmy2−cmz2−dku2+mb≤−θVX+mb=−θVX−mbθ=−θVX−L2<0.
Thus, we have(20)VXt−L2≤VXt0−L2e−θt−t0,and
(21)lim¯t→+∞VXt≤L2.which clearly shows that Ψ=X∣VX≤L2 is the globally exponential attractive set of system (2).The proof is complete.Remark 3.
(i)
We can figure out that the volume of the setΨ in (15) is vΨ=π2L4/2mmd according to Theorem 2.


## 3. Conclusions

In this paper, we considered some dynamics of a new 4D financial chaos system. We obtained the ultimate boundedness and global exponential attractive sets of this system according to Lyapunov stability theory. MATLAB simulations show that the proposed method is effective. Characteristic time scales of the 4D financial system, the homoclinic orbits, and heteroclinic orbits of the 4D financial system will be taken into consideration in the future.



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