# Chaos Control and Anticontrol of the Output Duopoly Competing Evolution Model

Authors: Zhaoqing Li; Yongping Zhang; Tongqian Zhang
Journal: Discrete Dynamics in Nature and Society
 (2017)

---

## Abstract

In the process of social development, there are a lot of competitions and confrontations. Participants in these competitions and confrontations always have different interests and goals. In order to achieve their goals, the participants must consider the opponent’s strategy to adjust their own strategies to achieve the interests of the optimization. This is called game. Based on the definition and its stability of the passive system, the passive control items are designed to the output of the duopoly competition evolution model, and the efficacy of the control methods is shown by the Lyapunov indexes. Then, the optimal function control method is taken to carry on the chaotic anticontrol to the chaotic system, and the Lyapunov indexes illustrate the control result. At last, the chaotic game of the system is introduced by combining the chaos control and anticontrol.


---

## Body

## 1. Introduction

In 1980, Stutzer firstly revealed the chaos phenomenon in the Haavelmo economic growth equation [1]. Then, it has been recognized that the economic model that is based on the traditional economic theory also has inherent randomness, which makes the effects of intervention means taken by traditional ways such as the fiscal policy or the financial policy and other corresponding macrocontrol means very limited [2]. The impact and revelation of chaos theory on traditional economics have aroused the concern and research of many experts and scholars. In [3], the Cournot Nash equilibrium and the local stability were investigated based on a dynamical multiproduct Cournot duopoly model in discrete time. And the tangential bifurcation and intermittent chaos were reported by numerical simulations. In [4], the generation of chaos in economic models through exogenous shocks was investigated. And a rigorous proof of the existence of chaos in the perturbed model was given and the results shed light on a novel source of chaos in economic models. And the OGY method proposed by Ott et al. [5] has made the study of chaotic control of economic system more systematic and extensive. In [6], the OGY method was used to stabilize chaos in the macroeconomic dynamic model, and the result showed that the government can, in principle, stabilize an unstable Walrasian equilibrium in a short time by varying income tax rates or government expenditures. In [7], necessary and sufficient criteria are established to guarantee CMGPS for the fractional-order complex chaos and fractional-order real chaos based on the stability theory of fractional-order systems.It is well known that duopoly model is one of the most representative oligopoly models. Duopoly competition is the oligopoly competition between two companies; based on the actions of the competitor, each company will take measures to maximize profits. In economics, the Cournot model and the Bertrand model are classical theories of the oligarchy, and their various extended models and deformations are used to deal with many problems in the field of industrial economics. Based on Bertrand model, the oligopolistic equilibrium price of insurance market in two cases is studied [8], one without the supply capacity limit and the other under the supply capacity limitation. Complex dynamics are shown in the study of the Cournot model [9], such as bifurcation and chaos with logarithmic demand functions. Puu [10, 11] discussed the dynamical properties of the Cournot oligarchic model and indicated that it will not converge to Nash equilibrium unless it is a complex trajectory. And the dynamical properties of the bounded rational duopoly model are also studied in [12].As an extension of the Cournot duopoly model, the output duopoly competing evolution model originally developed by Agiza et al. [13] has also been further studied. Du et al. [14] used the linear stabilization method to control chaos and demonstrated a variety of results of economic chaos control. They also analyzed the effects of both the output speed adjustment and the cost function type on the results. The method can guide the unstable orbit to follow the fixed point in the area of the unstable point and then enter the stable area around the fixed point within a given period of time. The limiter method is also applied to control the chaos of the output duopoly competing evolution model [15]. The limiter method realized the control of chaos and hyperchaos by limiting the strange attractor space in chaos and hyperchaotic systems (the hyperchaotic phenomenon occurs in the nonlinear high-dimensional system, that is, at least four-dimensional nonlinear system, with multidirectional instability, while the divergence of the trajectory occurs in only one direction in chaos system [16]).The output duopoly competing evolution model will be introduced to analysis in this paper. Boulding and Jarrett [17] proposed that the economic system has the property of the dissipative structure; namely, the economic system is open and moving away from the balance. Passive system, as a special case of dissipative system, is developed from network theory and other physical branches, and it carries out research from the energy properties of the system. It is a kind of dynamic system considering the energy exchange between the system and the outside world and can maintain the internal stability. For the system with interference, in order to stabilize the system, the feedback controller can be constructed based on the passive theory, so that the closed-loop response system can remain internally stable. The system controller designed based on the passive control theory can realize the global stability of the system without singularity problem, and it has strong robustness to the change of the system parameters and the external perturbation. It is a kind of essential nonlinear control method. Because the economic system has the dissipative structure property, in this paper, we introduce the passive control method in Section 5 to conduct the chaos control to the model; the global stability of the system is achieved by constructing passive control item based on passive theory, and the effectiveness of the results can be showed by numerical simulation. In addition, in Section 6, we consider the chaotic anticontrol of the output duopoly competing evolution model on the basis of passive control, where the method used is the optimal function method [18], and its effectiveness is also validated through the numerical simulation.The Cournot duopoly model embodies the idea of the game; that is, there are two enterprises to carry on the game of production, and each enterprise determines its own decisions based on the others’ production decisions. But this game is a static game; two oligarchs make decisions at the same time, and there is no room for reaction. In [19], a dynamic duopoly game with heterogeneous players was discussed. The paper analyzed how the limiter affects the dynamics of output and the performance of players and explored the number of the equilibrium points and the distribution of conditioned equilibrium points of the model. In [20], a nonlinear Kopel duopoly game with heterogeneous players is presented to show bifurcation diagrams, phase portraits, and sensitive dependence on initial conditions by numerical simulations. In [21], the chaotic behavior of the duopoly Cournot-Puu model was controlled by introduced delayed feedback control (DFC) method and the adaptive control method to the system, and chaos was controlled during the output adjustment process in the actual economic sense. In [22], two different time delay structures for the dynamical Cournot game with two heterogeneous players were researched; the dynamics of both players output adjustments are analyzed and simulated. The paper pointed that the system may lose stability due to either a period-doubling bifurcation or a Neimark-Sacker bifurcation, but the chaotic behaviors can be stabilized by the time-delayed feedback control. In [23], a cubic utility function that was derived from a constant elasticity of substitution production function (CES) was introduced, and, based on this, a two-dimensional Cournot duopoly game with horizontal product differentiation was modeled. The stability analysis of Nash point was considered and some numerical simulations were carried out to confirm the chaotic behavior of Nash equilibrium point. Differently from the fact that authors put importance on the analysis of chaotic behavior and chaos control, in this paper, in addition to chaos control, chaos anticontrol is also considered. And considering that the static game can not respond in a timely manner, the emergence of differential game solves this problem; that is, one first makes the decision and the other determines the output according to the previous decision information and the current decision, making decision-making more effective. In this paper, this idea is applied to the chaotic game of the output duopoly competing evolution model in Sections 3 and 7.Based on the output duopoly competing evolution model, this paper is divided into the following parts to discuss. Firstly, the fundamental theories are given in Section2 and the basic thought of chaotic game is introduced in Section 3. Secondly, the duopoly model with its fixed point analysis is introduced in Section 4. Then, Sections 5 and 6 discuss the chaos control and the antichaos control of the system, respectively. And, finally, we consider the chaotic game of the model based on Sections 5 and 6.


## 2. Fundamental Theory

Supposef:C→C is a map on the complex plane. If f(ω)=ω, ω is called the fixed point of f [21].The nonlinear generalized discrete system can be written as follows [22]:(1)Exk+1=Axk+fxk+Bωk,zk=Cxk+Dωk,where x(k)∈Rn, ω(k)∈Rp, and z(k)∈Rp represent the state, disturbed input, and output of the system, respectively; f(x(k)) is the nonlinear part of the system; E,A∈Rn×n, rank(E)=r<n (rank(T) means the ranks of the matrix T); B, C, and D are all matrixes with appropriate dimension.Suppose(E,A) is regular, causal, and stable [23], so the Lyapunov equation [23] is(2)ATVA-ETVE=-ETWE.And for every W>0 (W>0 means that W is positive definite), we have the only positive semidefinite solution V which satisfies the equation rankETVE=rank(E)=r.Definition 1 (see [22]).
For system (1), if there is a nonnegative definite function Vxk meeting the following inequality for any k≥0 and any input item ω(k),(3)Vxk+1-Vxk≤zTωk,then system (1) is called to be passive.
Without the input and output, system (1) is(4)Exk+1=Axk+fxk.For system (4), it is assumed that for any satisfying compatible initial condition Ex(0)=Ex0, the system has only one solution x(k).Consider the following nonlinear discrete controlled system(5)Exk+1=Axk+fxk+B1uk,where u(k)∈Rm is the control item and B1 is the arbitrary matrix with appropriate dimension.The aim is looking for a passive state feedback controlleru(k)=Gx(k) which can make the system(6)Exk+1=A+B1Gxk+fxkstable.According to the Lyapunov stability theory, (6) is E-asymptotically stable [24]; that is, limk→0⁡Exk=0; then it can be proved that limk→0⁡xk=0 [23], so, we can say that (6) has asymptotic stability of zero solution and is a passive system.Lemma 2 (see [22]).
For anyn dimensional vectors x and y, there is 2xTy≤xTx+yTy.There are results as follows.Theorem 3 (see [22]).
If system (4) meets the following conditions:(1)
fxk≤ρExk, ρ>0 is a constant;(2)
Ax(k)≤γExk, γ>0 is a constant;(3)
W-2γρλmax(V)I-ρ2λmaxI>0, where W, V are matrixes that meet (2) and λmax(M) is the biggest eigenvalue of M, then system (4) has asymptotic stability of zero solution.Theorem 4 (see [22]).
Suppose there is a constantε>0 which can make system (5) have(1)
fxk≤ρExk, ρ>0 is a constant;(2)
Ax(k)≤γExk, γ>0 is a constant;(3)
S<0, where(7)S=-W+2γρλmaxVI+ρ2λmaxI+ε2γ24λmaxVB1B1TV2+VB1B1TVI-
ε
γ
2
λ
m
a
x
V
B
1
B
1
T
V
I, and W, V are matrixes that meet (2); λmax(M) is the biggest eigenvalue of M.Then system (5) has a passive state feedback controller that can make (6) have asymptotic stability of zero solution and be passive. And the controller is (8)uk=-ε2B1TVAxk.The basic characteristic of chaos movement is that the movement is extremely sensitive to initial conditions [24]. And Lyapunov index is the quantitative index that can describe this phenomenon. For one dimensional mapping(9)xn+1=fxn.Consider the initial value x0 and the adjacent value x0+δx0. After one iteration, the distance between two points is (10)δx1=fx0+δx0-fx0=dfx0dxδx0.After n iterations, the distance between two points is (11)δxn=fnx0+δx0-fnx0=dfnx0dxδx0=eLE∗nδx0.Take n→∞; there is (12)LE=limn→∞⁡1nln⁡δxnδx0=limn→∞⁡1nln⁡dfnx0dxδx0.LE represents the separated index caused by each iteration on average of the distance between two points.


## 3. Basic Thought of Chaotic Game

Considerm-dimensional discrete chaotic system(13)wn+1=Gwn,where wn=x1n,x2n,…,xmnT is the state of the system at stage n. When parameters of (13) are within a certain range, (13) exhibits chaotic behavior. Chaos control is to make system stable, while anticontrol is to enhance chaos.In the following parts, we will discuss the chaos control and anticontrol. And then, on the basis of the theory of differential game, we will discuss the chaos game of system (13).Consider(14)wn+1=Gwn+pn+qn,where pn=pwn,p1,q1,…,pn-1,qn-1, qn=qwn,p1,q1,…,pn-1,qn-1 are controllers needing to be designed. pn is for stability and qn for chaos. Specific steps are as follows.Whenn=1, w2=Gw1+p1+q1. p1=pw1 is the first item that we need to design for stability, which is related to w1. q1=q(w1,p1) is the first item we will design for chaos after we got p1=p(w1).Whenn=2, w3=G(w2)+p2+q2. We can know that w3 is associated with G(w2), while G(w2) is associated with p1, q1, so the selection of p2 is related to p1, q1, and the selection of q2 needs to consider p1, q1, and p2.For then-1 step, wn=Gwn-1+pn-1+qn-1. wn is associated with G(wn-1), and G(wn-1) is associated with pn-2, qn-2. Considering stability and chaos game, the selection of pn-2 is related to pn-3, qn-3, and the selection of qn-2 needs to consider pn-3, qn-3, and pn-2. Therefore, pn-1 is related to p1, q1,…,pn-2, qn-2, and qn-1 is related to p1, q1,…,pn-2, qn-2, pn-1.Based on the above ideas, the selection of control items in the game will be discussed as follows.


## 4. The Output Duopoly Competing Evolution Model with Its Fixed Point Analysis

Consider the model as follows [13]:(15)q1t+1=q1t+α1q1ta-bq1t+q2t-b+2e1q1t-d11-s,q2t+1=q2t+α2q2ta-bq1t+q2t-b+2e2q2t-d21-s,where q1(t) and q2(t) are the output of two companies at time t, respectively, t=1,2,…; a,b are positive constants, and a is the highest price of the market; assume that the cost functions are C(q1)=c1+d1q1+e1q12 and C(q2)=c2+d2q2+e2q22, and the first derivatives of C(q1) and C(q2) are positive and the second derivatives are negative, so we can get di,ei>0; since companies’ marginal profits are demanded to be less than the price of the same product in market, we have d1+2e1x<a,d2+2e2y<a; αi is positive parameters, which shows the company’s adjustment speed.Let(16)Qq1,q2=q1+α1q1a-bq1+q2-b+2e1q1-d11-s,Rq1,q2=q2+α2q2a-bq1+q2-b+2e2q2-d21-s,assuming that the fixed point of model is q1∗,q2∗; according to definition, we have (17)Qq1∗,q2∗=q1∗,Rq1∗,q2∗=q2∗.The equations’ solutions have the following several cases:(1)
q1∗,q2∗=0,0;(2)
q1∗,q2∗=0,a-d2/2b+e2;(3)
q1∗,q2∗=a-d1/2b+e1,0;(4)
q1∗,q2∗=(a-d12b+2e2-ba-d2/3b2+4be1+4be2+4e1e2,  a-d22b+2e1-ba-d1/3b2+4be1+4be2+4e1e2).In this article, we just consider the fixed point that bothq1∗ and q2∗ are not zero.


## 5. Chaos Control of the Output Duopoly Competing Evolution Model

In order to facilitate the following analysis, the system (15) needs to be rewritten into the form of system (4). If we directly change system (15) into (4), there is (18)E0xk+1=A0xk+f0xk,where E0=1001, A0=a11a12a21a22, f0xk=f1xkf2xk, a11,a12,a21,a22,f1xk, and f2xk will be given in the following discussion. It is quite clear that E0 cannot meet the requirement that rank(E)=r<n; here, n=2. So, a function q3(t) is tried to be introduced into system (15), which meets the following equation:(19)0∗q3t+1=q3t∗means  multiplication.We have variable substitution (20)x∗=q1∗;y∗=q2∗,x1t=q1t-x∗;x2t=q2t-y∗;x3t=q3t.So, system (15) can be written as(21)Exk+1=Axk+fxk,where (22)E=100010000,xk=x1kx2kx3k,A=a11a120a21a220001,fxk=f1xkf2xkf3xk,a11=1+α1a1-s-bα11-sx∗+y∗-aα1b+2e11-sx∗-α1d11-s-α1x∗b1-s;a12=-α1bx∗1-s;a21=-α2by∗1-s;a22=1+α2a1-s-bα21-sx∗+y∗-aα2b+2e21-sx∗-α2d21-s-α2x∗b1-s;f1xk=-α1b1-sx1kx2k+α1b1-s+α1b+2e11-sx12t+α1b1-sx∗x∗+y∗+α1b+2e11-sx∗2+α1d11-sx∗;f2xk=-α2b1-sx1kx2k+α2b1-s+α2b+2e21-sx22t+α2b1-sy∗x∗+y∗+α2b+2e21-sy∗2+α2d21-sy∗;f3xk=0.Here,rank(E)=r<n, where rank(E)=2<n=3. Therefore, the fixed point of system (21) is (0,0,0).According to the content of passive control in the basic theory, we can get the following theorem.Theorem 5.
Suppose there is a constantε>0 which makes system (5) have(1)
fxk≤ρExk, ρ>0 is a constant;(2)
Ax(k)≤γExk, γ>0 is a constant;(3)
S<0, where(23)S=-W+2γρλmaxVI+ρ2λmaxI+ε24λmaxVV2+VVI-εγ2λmaxVVI-εργλmaxVVI;and W, V are matrixes that meet (2); λmax(M) is the biggest eigenvalue of M.
Then system (5) has a passive state feedback controller that can make (6) have stability of zero solutions and be passive. And the controller is (24)uk=-ε2B1TVAxk.According to conditions of Theorem 5, a control item is introduced to (21) (25)uk=-ε2B1TVAxk.Due to the arbitrariness of matrix B1, let B1=I. So the controlled system is(26)Exk+1=Axk+fxk-ε2VAxk.We will prove that the system (26) has stability of zero solutions and (26) is passive.Proof.
The generalized Lyapunov function of (6) is selected as VExk=ExkTVExk. And according to the conditions of the Lemma and Theorem 5, there is(27)ΔVExk=VExk+1-VExk=A+Gxk+fxkTVA+Gxk+fxk-ExkTVExk=-ExkTWExk+2AxkTVGxk+2AxkTVf+GxkTVGxk+fxkTVfxk.
According to (16), we have G=-ε/2VA, so(28)ΔVExk≤ExkT-W+2γρλmaxVI+ρ2λmaxI+ε24λmaxVTVVI-εγ2λmaxVVI-εργλmaxVVI.
Let(29)S=-W+2γρλmaxVI+ρ2λmaxI+ε24λmaxVTVVI-εγ2λmaxVVI-εργλmaxVVI.So(30)ΔVExk≤ExkTSExk.
According to Theorems3, and 4 and the conditions of Theorem 5, (5) has stability of zero solutions.
Because the model does not have any input and output except the control itemu(k), and ω(k) is arbitrary, so we choose ω(k)=0. Then we just need to prove (31)VExk+1-VExk≤0.
The generalized Lyapunov function described above is the nonnegative definite function required in the definition. Thus, according to the conditions of the theorem and the proof above, we have(32)VExk+1-VExk≤0.
Therefore, system (6) is passive.For example, we take the parameters in (15) as a=10, b=1, d1=1, d2=1, e1=1, e2=1.1, s=0.3, α1=0.52, and α2=0.1. Then the fixed point is x∗,y∗=1.8228,1.7089. For these parameters, we can get Lyapunov indexes of (15) as 0.2540 and -1.4579. Because 0.2540>0, system (15) under parameters above is chaotic.Under the parameters, we have,(33)E=100010000,A=-1.654-0.66350-0.11960.49760001,fxk=f1xkf2xkf3xk,where f1xk=-0.364x1kx2k1.456x12t+6.2281, f2xk=-[0.07x1(k)x2(k)+0.294x22(t)+1.1963], f3xk=0, and the fixed point is (0,0,0).WhenS0≠4.276 and S0≠1.63, we can get |S0E-A|≠0; then we can say that (E,A) is regular [25].According to the fundamental theory,W is arbitrary and we take W as(34)W=010110001;bring A, E, and W into (2); then we have the only positive semidefinite solution(35)V=0.8473-0.11470-0.11470.47240000.In an actual market of oligarch competition, the output of participates in the competition is not infinite because the economic resources are always limited. Therefore, we assume that production is bounded. Then there is anM>0 satisfying x1k<M, x2k<M, x3k<M.And(36)Exk=x12k+x22k,Axk=1.654x1k+0.6635x2k2+-0.1196x1k+0.4976x2k2+x32k,fxk=0.364x1tx2t+1.456x12t+6.22812+0.07x1tx2t+0.294x22t+1.19632.Because production is limited, we can getA(x(k))≤γEx(k) when γ>1.3594 and when ρ is big enough, we have ρ2M>3.4427M4+23.5337M2+40.2203.Then the controlled system is(37)x1t+1=-1.654x1t-0.6635x2t-0.364x1tx2t+1.456x12t+6.2281+ε21.3877x1t+0.6193x2t,x2t+1=-0.1196x1t+0.4976x2t-0.07x1tx2t+0.294x22t+1.1963-ε20.1332x1t+0.3112x2t.The bifurcation figure is given with the change of the parameterα. It can be seen from the Figure 1 that when α is taken as 0.476, the system output bifurcation occurs. When α is 0.5, the system output shows obvious chaos. Then the output change of the system when α is 0.52 will be given.Figure 1
The bifurcation of the output with the change ofα.The blue curve represents the output ofx, and the red one represents y.It can be seen from Figure2 that, under the given parameters in this example, the output of the system exhibits an irregular increase or decrease with the number of iterations increases; that is, the fixed point of the system (15) is unstable, and the Lyapunov indexes of the system can also show this.Figure 2
Original output of system.It can be seen from Figure3 that, after the introduction of passive controller, with the change of control coefficient, the output change gradually becomes regular under the increase of the iterations, and finally the system achieves global stability. The output of the system is stabilized to the fixed point, which means that the enterprises’ output is no longer unpredictable and confusing. With this control item, the output trends to be stable, and enterprises can control their output to avoid wasting of resource. As can be seen from the figure, with the increase of the control coefficient, the output stabilizes to the fixed point faster and faster. In Figure 3(c), the control system needs about 50 iterations to stabilize to the fixed point, while, in Figure 3(d), the system just needs 25 iterations.Figure 3
The changing of the output of the controlled system (a)ε=0.8; (b) ε=1; (c) ε=1.2; (d) ε=1.4.
(a)
(b)
(c)
(d)Under the selected parameters above, the corresponding Lyapunov indexes of control system are as follows(1)
ε=0.8, Lyapunov indexes are 0.3477 and -0.4531;(2)
ε=1, Lyapunov indexes are 0.0421 and -0.1474;(3)
ε=1.2, Lyapunov indexes are -0.0526 and -0.0527;(4)
ε=1.4, Lyapunov indexes are -0.0526 and -0.0527.The change of Lyapunov indexes of the control system withε is shown in Figure 4. The red curve represents the Lyapunov index of x, and the green one represents y.Figure 4
The change of Lyapunov indexes withε.Figure4 shows the change of Lyapunov indexes with ε, and, with the change of ε, the Lyapunov index of x is getting smaller and y’s is getting bigger, but finally the Lyapunov indexes of two variables are negative. This shows the effectiveness of passive control.Figure5 shows the system output with the change of the control parameter ε. It can be seen from Figure 5 when ε is bigger than 1 that the output is no longer showing the case of bifurcation but is stable at the same value. The results of Figures 5 and 4 are in agreement, and this shows the effectiveness of the control.Figure 5
The bifurcation of the output with the change ofε.


## 6. The Chaos Anticontrol of the Output Duopoly Competing Evolution Model

Zero solutions of system (26) are stable and the passivity has been proved. Now the optimal function control method [17] is taken to conduct the chaos anticontrol of (26). Equation (26) can be written in the following form, and we just consider x1 and x2:(38)x1n+1=x1n+Fx1n,x2n,x3n,x2n+1=x2n+Gx1n,x2n,x3n,where(39)Fx1n,x2n,x3n=a11x1n+a12x2n+f1x1n,x2n-ε2a11V11+a21V12x1n+a12V11+a22V12x2n+V13x3n,Gx1n,x2n,x3n=a21x1n+a22x2n+f2x1n,x2n-ε2a11V21+a21V22x1n+a12V21+a22V22x2n+V23x3n,f1x1n,x2n=-α1b1-sx1nx2n+α1b1-s+α1b+2e11-sx12n+α1b1-sx∗x∗+y∗+α1b+2e11-sx∗2+α1d11-sx∗,f2x1n,x2n=-α2b1-sx1nx2n+α2b1-s+α2b+2e21-sx22n+α2b1-sy∗x∗+y∗+α2b+2e21-sy∗2+α2d21-sy∗,and a11, a12, a21, and a22 have been given in the fifth part, and V11,V12,V21,V22 are the corresponding elements of V.We consider the following anticontrol system:(40)x1n+1=x1n+Fx1n,x2n,x3n+kx1n+Fx1n,x2n,x3n-x1n,x2n+1=x2n+Gx1n,x2n,x3n+kx2n+Gx1n,x2n,x3n-x2n.For example, we take the parameters in (40) as a=10, b=1, d1=1, d2=1, e1=1, e2=1.1, s=0.3  α1=0.52, and α2=0.1. According to the previous example, we can get the corresponding anticontrol system(41)x1n+1=x1n-k+12.654x1n+0.6635x2n+0.364x1nx2n+1.456x12n+6.2281-ε21.3877x1n+0.6193x2n,(42)x2n+1=x2n-k+11.1196x1n-0.4976x2n+0.07x1nx2n+0.294x22n+1.1963+ε20.1332x1n+0.3112x2n,where ε=1.1.The change of the output with the change of steps under the parameters is shown in Figure6.Figure 6
Output withε=1.1.The Lyapunov indexes of Figure6 are -0.0526, -0.0527; that is to say, system is stable.In (40), the changes of steps with k can be seen in Figure 7.Figure 7
The changes of the output with antichaos control item (a)k=0.3; (b) k=0.35; (c) k=0.4; (d) k=0.45.
(a)
(b)
(c)
(d)It can be seen from Figure7 that the output of the system has obvious irregular variation for the passive system with optimal function control item, and the output of the system becomes more complicated with the increase of the control coefficient. At this point the fixed point becomes unstable fixed point again, and the system presents chaos. The results are also demonstrated by the following Lyapunov indexes.Under the selected parameters above, the corresponding Lyapunov indexes of control system are as follows:(1)
k=0.3, Lyapunov indexes are 0.5499 and -0.6553;(2)
k=0.35, Lyapunov indexes are 0.5343 and -0.6396;(3)
k=0.4, Lyapunov indexes are 0.3188 and -0.4242;(4)
k=0.45, Lyapunov indexes are 0.5348 and -0.6402.The change of Lyapunov indexes of the anticontrol system withk is shown in Figure 8.Figure 8
The change of Lyapunov exponents withk.It can be shown in Figure8 that with the anticontrol item, the Lyapunov exponent of x is always positive, indicating that the system is chaotic.Finally we give the bifurcation figure with the change of the parameterk. It can be seen from the Figure 9 that when the anticontrol parameter k is taken as 0.15, the system output bifurcation occurs. When k is 0.42, the system output shows obvious chaos. The original stability system becomes a chaotic system with the change of parameter k.Figure 9
The bifurcation of the output with the change of anticontrol parameterk.


## 7. Discussion of the Chaotic Game

For the output duopoly competing evolution model, we consider(43)zn+1=Hzn+μzn+vzn,where z(n)=x1(n)x2(n)x3(n), and Hzn is obtained since antichaos control is conducted on system (26), and (44)Hzn=x1n+Fx1n,x2n,x3n+kx1n+Fx1n,x2n,x3n-x1nx2n+Gx1n,x2n,x3n+kx2n+Gx1n,x2n,x3n-x2nx3n-kε2a11V31+a21V32x1n+a12V31+a22V32x2n+V33x3nF, G have been given in the sixth part and a11, a12, a21, and a22 are in the fourth part. V11, V12, V21, and V22 are the corresponding elements of V. μ(z(n)) is for stability and v(z(n)) is for chaos.According to basic thought, there is the following process.Whenn=1,(45)z2=Hz1+μ1+v1,without control items μ1 and v1, we have z(2)=Hz1, which can be rewritten as the standard form similar to (4); that is,(46)E1z2=A1z1+lz1,where(47)E1=100010000,A1=1+k+1a11-ε2a11V11+a21V12k+1a12-ε2a12V11+a22V12-ε2k+1V13k+1a21-ε2a11V21+a21V221+k+1a22-ε2a12V21+a22V22-ε2k+1V23-εk+12a11V31+a21V32-εk+12a12V31+a22V321-εk+12V33,lz1=fx1, and f(x(k)) has been given in the second part.Then the methods above are used to study this problem. WhenE1z(2)=A1z(1)+lz1+μ1 satisfies the conditions of Theorem 5, there is a state feedback passive controller μ1=-ε1/2V1A1z(1), making z(2)=Hz1+μ1 stable, where ε1 is a constant bigger than zero and V1 is the only positive semidefinite solution satisfying A1TV1A1-E1TV1E1=-E1TWE1.Then we considerv1. Using the method of anticontrol, we have (48)v1=x11+1+kFx11,x21,x31+k1x11+1+kFx11,x21,x31-x11x21+1+kGx11,x21,x31+k1x21+1+kGx11,x21,x31-x21x31-kε2a11V31+a21V32x11+a12V31+a22V32x21+V33x31+Mx11,x21,x31,,where M(x1(1),x2(1),x3(1))=-k1ε1/2[(a11′V31′+a21′V32′)x1(1)+(a12′V31′+a22′V32′)x2(1)+V33′x3(1)], a11′=1+(k1+1)[a11-ε/2(a11V11+a21V12)], a21′=k1+1a21-ε/2a11V21+a21V22, a12′=k1+1a12-ε/2a12V11+a22V12, a22′=1+k1+1a22-ε/2a12V21+a22V22. V31′, V32′, and V33′ are corresponding elements of V1, because v1 is related to z(1) and M(x1(1),x2(1),x3(1)). And M(x1(1),x2(1),x3(1)) is related to μ1, so v1 is the anticontrol item related to μ1. That is to say, μ1 and v1 can enable system (45) to realize the chaos game.Whenn=2, z(3)=H(z(2))+μ2+v2. Similar to the above analysis, firstly, we consider z(3)=H(z(2)) without control items μ2, v2, which can also be rewritten as the standard form like E1z(2)=A1z(1)+l(z(1)); then we can get the passive control item like μ2=-ε2/2V2A2z(2). Because of z(2)=H(z(1))+μ1+v1, the choose of μ2 is related to μ1, v1. According to the analysis of n=1, we can get the anticontrol item like v1, and v2 is connected with z(2) and μ2. Thus v2 is related to μ1, v1, and μ2.By similar analysis for thenth step, for z(n+1)=H(z(n))+μn+vn, we can get the passive control item like μn=-εn/2VnAnz(n). And according to the above analysis, μn is related to μ1, v1, μ2, v2,…,μn-1, vn-1. Furthermore, we get the anticontrol item vn related to μ1, v1, μ2, v2,…, μn-1, vn-1, μn.


## 8. Conclusion

With the rapid development of social economy, science, and technology, and the evolution of the complex economic system, the traditional theory of economics based on linear system theory is faced with great challenge. Chaos theory mainly studies the influence of “instability,” “structural change,” on the system, and the causes of “abnormal behavior,” which contrasts with the “stable” and “reasonable behavior” that traditional economic theory emphasizes. The traditional economy advocates certainty and balance, believing that the economic system will remain in a quiescent state without external factors and emphasizes the linear response. And chaotic economists proceed with the “imperfect and unbalanced world” to explain the economic system, which is full of fuzziness, chaos, and contingency, and try to find a dialectical relationship of the transition between order and disorder [26]. Therefore, chaos theory opens up a new way for the study of economic chaos behavior, and, with chaos theory, it may reveal the complex dynamic characteristics that can not be explained by traditional methods [2, 27–30]. As a new hot-spot and the highlight in chaos research and application, chaos control has received extensive attention, and it has also been applied to the study of chaos in economics. Competitions, whether in the military or economic markets, are real and always exist. How to use low investment to realize the maximal interests has been the basis that provides a chance for practical uses of game thoughts.The chaos game of the output duopoly competing evolution model under the given parameters is discussed in this paper. Firstly, we use the passive theory to carry out chaos control. Then, we have chaos anticontrol for the stable system, and the results can be shown by the Lyapunov exponents. And finally, the chaos game about the system is introduced.



---

*Source: 4121635-2017-11-06.xml*