# Computation of the Stability and Complexity about Triopoly Price Game Model with Delay Decision

Authors: Yuling Wang; Shubing Guo; Shide Duan
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We develop the price game model based on the entropy theory and chaos theory, considering the three enterprises are bounded rationality and using the cost function under the resource constraints; that is, the yield increase will bring increased costs. The enterprises of new model adopt the delay decision with the delay parametersτ1 and τ2, respectively. According to the change of delay parameters τ1 and τ2, the bifurcation, stability, and chaos of the system are discussed, and the change of entropy when the system is far away from equilibrium is considered. Prices and profits are found to lose stability and the evolution of the system tends to the equilibrium state of maximum entropy. And it has a big fluctuation with the increase of τ1 and τ2. In the end, the chaos is controlled effectively. The entropy of the system decreases, and the interior reverts to order. The results of this study are of great significance for avoiding the chaos when the enterprises make price decisions.


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## Body

## 1. Introduction

The oligopoly is a universal market state between perfect competition and complete monopoly. Game theory, entropy theory, and nonlinear dynamics provide new impetus for oligopoly theory. There are a lot of oligopolies in the market, such as China Mobile, China Unicom, and China Telecom, forming a complex system with increasing entropy. These oligopoly enterprises constantly carry on the price game in order to maximize the benefits. Many scholars have studied the content of oligopoly game from different perspectives, such as entropy theory, chaos, and game theory. Zhang et al. [1] built a Bertrand repeated game model with linear demand function and studied its system complexity. Xu and Ma [2] investigated the dynamic model of a Bertrand game with delay in insurance market. They discussed the existence of the Nash equilibrium point of the game and researched the stability of the system. Sun and Ma [3] considered a two-player quantum game in the presence of a thermal decoherence modeled with the method of a rigorous Davies. It shows how the energy dissipation and pure decoherence make changes on the payoffs of the players in the game. Dajka et al. [4] studied the complex dynamics of a nonlinear model on the basis of Bertrand game in Chinese cold rolled steel market. Fanti et al. [5] analyzed the dynamics of a Bertrand duopoly with products which become divided. The results showed that an increase in either the degree of substitutability or complementarity between products of different varieties was the reason of complexity in a competition game. Xiangyu and Xiaoyong [6] used the information theory and entropy theory to build the models to measure the entropy of the four market structures which are perfect competition, monopolistic competition, oligopoly, and complete monopoly and compare the entropy of the four market structures. Naimzada and Tramontana [7] considered a Cournot–Bertrand duopoly model based on linear demand and cost functions with product differentiation. Li and Ma [8] considered the R&D input competition model in oligopoly market on the basis of that the players are heterogeneous, bounded rational, and adaptive adjustment. Fan et al. [9] investigated two types of players and concluded the output duopoly game with heterogeneous players. They studied the influence of players’ different behavior on the dynamics of game. Yali [10] built a duopoly game model and investigated its stability with bounded rationality strategy and state delay. Gao et al. [11] discussed equilibrium stability of a nonlinear Cournot duopoly game, where one player can evaluate its opponent’s output in the future in light of straightforward extrapolative foresight. Peng et al. [12] analyzed a dynamic of triopoly Bertrand repeated model with the zero marginal cost. Bischi and Naimzada [13] concluded the dynamical characteristics of bounded rationality duopoly game. Ma and Tu [14] carried out the corresponding extension of the complex dynamics to macroeconomic model with time delays considering the macroeconomic model of money supply. Ma and Wang [15] considered a closed-loop supply chain with product recovery, which is composed of one manufacturer and one retailer. The situation may lead to complicated dynamic phenomena such as bifurcation and chaos. That is to say, the entropy of the system is increasing too. Hale [16] investigated existence and the local stable region of the Nash equilibrium point. Ma and Si [17] studied a continuous Bertrand duopoly game model with two-stage delay. Ma and Bangura [18] studied financial and economic system when the three parameters were changed.By combining them, it is found that most of the studies are based on the discrete system, and the attention to the research of continuous system is not much, with lack of analysis from the in-system state and entropy theory, considering the delayed decision is less. Therefore, the model of [19] is improved based on the entropy theory and chaos theory, considering the three companies are bounded rationality and using a new cost function, and its chaotic characteristics and system entropy changes were analyzed. In the course of the study, the special case of τ1=τ2=τ is overcome, τ1≠τ2, and τ1>0, τ2>0 are discussed. The improved model is more fit to the reality, and the research results are of guiding significance to the enterprise price decision.This paper is organized as follows: in Section2, based on [19], a triopoly price game model with delay is improved. In Section 3, the stability of system and the existence of Hopf bifurcation are analyzed. In Section 4, numerical simulation is used to find out the influence of delay on the stability of price and profit by virtue of time series, the attractor, bifurcation diagram, Lyapunov exponent, 3D surface chat, and initial value sensitivity, as well as the contacts between dynamic state and the situation of entropy change in the system. In Section 5, the effective control of chaos is achieved by control method of the state variables feedback and parameter variation in the system. Finally, we have some conclusions in the last section.


## 2. The Model

The triopoly dynamic game model is developed in [19] which makes adaptive decision, bounded rational decision, and delayed bounded rational decision, respectively. The stability of the system and the existence of Hopf bifurcation are studied in this paper. The model is described as follows:(1)p˙1t=v1p1a+d1wp2t+d11-wp2t-τ+f1p3t+b1c12b1-p1t-τ,p˙2t=v2p2a-2b2wp2t-2b21-wp2t-τ+d2p3t+f2p1t+b2c2,p˙3t=v3p3a-2b3p3t+d3p1t+f3wp2t+f31-wp2t-τ+b3c3,where a,bi,di,fi>0, i=1,2,3, a represents the largest market demand for products, bi is elastic demand, di, fi represents the substitution rate between the two companies, respectively, pi, qi denote the price and output of the product, respectively, 0<w<1 represents the weight of the current price, and 1-w represents the weight of price of t-τ time. The cost function with linear form is Ci(qi)=ciqi, i=1,2,3, and ci is marginal profit. In (1), the first enterprise adopts the adaptive pricing strategy with delay, where τ1 denotes the delay parameter; the other two enterprises employ the finite rational pricing strategy. In addition, the second enterprise used the postponement strategy, where τ2 stands for the delay parameter. The linear cost function under the condition of sufficient resources was used. Then τ1=τ2=τ was discussed.Because price information is asymmetry, we consider three companies are bounded rationality based on model [19] and build the price game model with enterprises 1 and 2 with delay parameters τ1and τ2, respectively. The cost function will obviously increase under limited resources; that is, Ci(qi)=ci0+ciqi2i=1,2,3, where ci0 is the fixed cost. We further have the improved model with price game:(2)p˙1t=ν1p11+2b1c1a-2b1+2b12c1p1t-τ1+d1+2b1c1d1p2t-τ2+f1+2b1c1f1p3t,p˙2t=ν2p21+2b2c2a-2b2+2b22c2p2t-τ2+d2+2b2c2d2p3t+f2+2b2c2f2p1t-τ1,p˙3t=ν3p31+2b3c3a-2b3+2b32c3p3t+d3+2b3c3d3p1t-τ1+f3+2b3c3f3p2t-τ2.


## 3. Local Stability at Equilibrium Points

In a competitive market, the equilibrium points must be nonnegative. Considering generality, we assume thatE∗(p1∗,p2∗,p3∗) is a Nash equilibrium point of model (2), where(3)p1∗>0,p2∗>0,p3∗>0.We study the existence of Hopf bifurcation of the system atE∗(p1∗,p2∗,p3∗). Let h1(t)=p1(t)-p1∗, h2(t)=p2(t)-p2∗, and h3(t)=p3(t)-p3∗, with p1(t), p2(t), and p3(t) instead of h1(t), h2(t), and h3(t), respectively, when h=0. We have the linear form of the system through Jacobian matrix as follows:(4)p1˙t=-v1p1∗2b1+2b12c1p1t-τ1+v1p1∗d1+2b1c1d1p2t-τ2+v1p1∗f1+2b1c1f1p3t,p2˙t=-v2p2∗2b2+2b22c2p2t-τ2+v2p2∗d2+2b2c2d2p3t+v2p2∗f2+2b2c2f2p1t-τ1,p3˙t=-v3p3∗2b3+2b32c3p3t+v3p3∗d3+2b3c3d3p1t-τ1+v3p3∗f3+2b3c3f3p2t-τ2.The determinant of (4) is(5)λE-JE∗=0,where(6)JE∗=J11J12J13J21J22J23J31J32J33,J11=-v1p1∗2b1+2b12c1,J12=v1p1∗d1+2b1c1d1,J13=v1p1∗f1+2b1c1f1,J21=v2p2∗f2+2b2c2f2,J22=-v2p2∗2b2+2b22c2,J23=v2p2∗d2+2b2c2d2,J31=v3p3∗d3+2b3c3d3,J32=v3p3∗f3+2b3c3f3,J33=-v3p3∗2b3+2b32c3.Therefore the characteristic equation for system (4) is(7)λ3+k1λ2+k2λ2+k3λe-λτ1+k4λ2+k5λe-λτ2+k6λ+k7e-λτ1e-λτ2=0,where(8)k1=2b3p3∗v3+2b32c3p3∗v3,k2=2b1p1∗v1+2b12c1p1∗v1,k3=4b1b3p1∗p3∗v1v3-d3f1p1∗p3∗v1v3+4b12b3c1p1∗p3∗v1v3+4b1b32c3p1∗p3∗v1v3+⋯-2b3c3d3f1p1∗p3∗v1v3,k4=2b2p2∗v2+2b22c2p2∗v2,k5=4b2b3p2∗p3∗v2v3-d2f3p2∗p3∗v2v3+4b22b3c2p2∗p3∗v2v3+4b2b32c3p2∗p3∗v2v3+⋯-2b3c3d2f3p2∗p3∗v2v3,k6=4b1b2p1∗p2∗v1v2+4b12b2c1p1∗p2∗v1v2+4b1b22c2p1∗p2∗v1v2+⋯-4b1b2c1c2d12f2p1∗p2∗v1v2,k7=8b1b2b3p1∗p2∗p3∗v1v2v3-2b1d2f3p1∗p2∗p3∗v1v2v3+⋯-8b1b2b3c1c2c3d12d2d3p1∗p2∗p3∗v1v2v3.We discuss the effects of τ1, τ2 on the stability of system (4) when τ1≠τ2, τ1>0, τ2>0.At this point, the characteristic equation of system (4) is(9)λ3+k1λ2+k2λ2+k3λe-λτ1+k4λ2+k5λe-λτ2+k6λ+k7e-λτ1+τ2=0.We consider (2) with τ2 in its stable range, regarding τ1 as a parameter. Taking into account the generality, we discuss system (2) under the case mentioned in [14], and τ2∈0,τ20. τ20 is defined as in [14]. Therefore we have (10)τ2kj=1ω2karccos⁡D-ACω2k4+AE-BDω2k2C2ω2k4+D2-2CEω2k2+E2+2jπω2k,k=1,2,3,4,5,j=0,1,2,…τ20=min⁡τ2k0,k∈1,2,3,4,5,ω20=ω2k0,where ω2k, i=1,2,3,…,k, are the positive roots of(11)H1ω210+H2ω28+H3ω26+H4ω24+H5ω22+H6=0,where(12)H1=C2,H2=D2-2CE-2BC2+A2C2-C4,H3=4BCE+E2+A2D2+B2C2+2D2B-2A2CE+4C3E-2C2D2,H4=A2E2-2E2B-2B2CE+B2D2-6C2E2-D4-4CED2,H5=B2E2-2D2E2+4CE3,H6=-E4,A=k1+k2,B=k3,C=k4,D=k5+k6,E=k7.Let λ=iω1(ω1>0) be a root of (9). Then (13)A4sin⁡ω1τ1+B4cos⁡ω1τ1=ω13,C4sin⁡ω1τ1+D4cos⁡ω1τ1=k1ω12+k4ω12cos⁡ω1τ2-k5ω1sin⁡ω1τ2,where(14)A4=k2ω12-k6ω1sin⁡ω1τ2-k7cos⁡ω1τ2,B4=k3ω1+k6ω1cos⁡ω1τ2-k7sin⁡ω1τ2,C4=k3ω1+k6ω1cos⁡ω1τ2-k7sin⁡ω1τ2,D4=k6ω1sin⁡ω1τ2-k2ω12+k7cos⁡ω1τ2.For (13), we can obtain that(15)sin⁡ω1τ1=N1ω14+N2ω13+N3ω12+N4ω1N10ω14+N11ω13+N12ω12+N13ω1+N14,cos⁡ω1τ1=N5ω15+N6ω14+N7ω13+N8ω12+N9ω1N10ω14+N11ω13+N12ω12+N13ω1+N14,N1=k6cos⁡ω1τ2-k2k4cos⁡ω1τ2-k1k2+k3,N2=k4k6cos⁡ω1τ2sin⁡ω1τ2+k1k6sin⁡ω1τ2-k7sin⁡ω1τ2+k2k5sin⁡ω1τ2,N3=k4k7cos2⁡ω1τ2+k1k7cos⁡ω1τ2-k5k6sin2⁡ω1τ2,N4=k5k7cos⁡ω1τ2sin⁡ω1τ2,N5=k2,N6=-k6sin⁡ω1τ2,N7=k1k3+k1k6cos⁡ω1τ2+k3k4cos⁡ω1τ2+k4k6cos2⁡ω1τ2-k7cos⁡ω1τ2,N8=k1k7sin⁡ω1τ2-k4k7cos⁡ω1τ2sin⁡ω1τ2+k5k6sin⁡ω1τ2cos⁡ω1τ2+k3k5sin⁡ω1τ2,N9=k5k7sin2⁡ω1τ2,N10=k22,N11=-2k2k6sin⁡ω1τ2,N12=k32+2k3k6cos⁡ω1τ2+k62-2k2k7cos⁡ω1τ2,N13=-2k3k7sin⁡ω1τ2,N14=k72.For (15), we obtain the following equation:(16)M10ω110+M9ω19+M8ω18+M7ω17+M6ω16+M5ω15+M4ω14+M3ω13+M2ω12+M1ω1+M0=0,M10=N52,M9=N12N62+2N5N7-N102,M8=2N5N6,M7=2N1N2+2N5N8+2N6N7-2N10N11,M6=N22+2N1N3+N72+2N6N8+2N5N9-N112-2N10N12,M5=2N1N4+2N2N3+2N7N8+2N6N9-2N10N13-2N11N12,M4=N32+2N2N4+N82+2N7N9-N122-2N11N13-2N10N14,M3=2N3N4+2N8N9-2N12N13-2N11N14,M2=N42+N92-N132-2N12N14,M1=-2N13N14,M0=-N142.Suppose that (H1): (16) has finite positive roots. We define the roots of (16) as ω11,ω12,ω13,…,ω1k. For every fixed ω1i (i=1,2,3,…,k), there exists a sequence τ1i(j)∣j=0,1,2,… which satisfies (16). It is(17)τ1ij=1ω1iarccos⁡N5ω1i5+N6ω1i4+N7ω1i3+N8ω1i2+N9ω1iN10ω1i4+N11ω1i3+N12ω1i2+N13ω1i+N14+2jπω1i,i=1,2,3,…,k,j=0,1,2,….Let(18)τ10=min⁡τ1ij∣i=1,2,3,…,k,j=0,1,2,…,ω10=ω1i0.When τ1=τ10, (9) has a pair of purely imaginary roots ±iω10 for τ2∈0,τ20. In the following, we take the derivative of λ with respect to τ1 in (9) for the transversality condition of Hopf bifurcation, and we have(19)dλdτ1-1=-3λ2-2k1λ-2k2λ+k3e-λτ1-2k4+k5e-λτ2-k6e-τ1+τ2k2λ3+k3λ2e-λτ1+k4λ3+k5λ2e-λτ2+k6λ+k7λe-λτ1+τ2-τ1λ.Then we further have(20)dλτ10dτ1λ=iω10-1=S1S3+S2S4S32+S42,where(21)S1=3ω102-2k2ω10sin⁡ω10τ10+k3cos⁡ω10τ10-2k4ω10sin⁡ω10τ2-k5cos⁡ω10τ2-k6cos⁡ω10τ10-ω10τ2,S2=k5sin⁡ω10τ2-2k1ω10-2k2ω10cos⁡ω10τ10-k3sin⁡ω10τ10-2k4ω10cos⁡ω10τ2-k6sin⁡ω10τ10+ω10τ2,S3=-k2ω103sin⁡ω10τ10-k3ω102cos⁡ω10τ10-k4ω103sin⁡ω10τ2-k5ω102cos⁡ω10τ2+k6+k7ω10sin⁡ω10τ10+ω10τ2,S4=-k2ω103cos⁡ω10τ10+k3ω102sin⁡ω10τ10-k4ω103cos⁡ω10τ2+k5ω102sin⁡ω10τ2+k6+k7ω10cos⁡ω10τ10+ω10τ2.Obviously, if (H2):S1S3+S2S4≠0, based on the above discussions and by the general Hopf bifurcation theorem in [15], we can obtain the results as follows.IfH(1)-H(2) hold, when τ2∈0,τ20, then the Nash equilibrium point E∗(p1∗,p2∗,p3∗) of system (2) is asymptotically stable for τ1∈0,τ10 and it is unstable as τ1>τ10. System (2) will be under Hopf bifurcation at E∗(p1∗,p2∗,p3∗) when τ1=τ10.


## 4. Numerical Simulations

The impacts of delay on the stability of system (2) are analyzed by a series of tools in this section. It supports the theoretical research in Section 3 by time series, bifurcation, Lyapunov exponents, attractor, and initial value sensitivity.The parameters of system (2) are taken to be a=5, b1=3.2, b2=3.5, b3=3.8, d1=0.3, d2=0.4, d3=0.5, f1=0.35, f2=0.45, and f3=0.55; the marginal costs of three dairy product companies are c1=0.003, c2=0.006, and c3=0.009; the initial prices of their products are p1(0)=0.4, p2(0)=0.5, and p3(0)=0.6; the speeds of price adjustment are v1=v2=v3=0.5; the fixed cost of the enterprise is c10=1, c20=1.5, and c30=2. Considering the following system, it is easy to calculate the Nash equilibrium point of system (2) which is E∗(0.8723,0.8329,0.8012).(22)p1˙t=0.205.0960-6.4614p1t-τ1+0.3058p2t-τ2+0.3567p3t,p2˙t=0.255.2100-7.1470p2t-τ2+0.4168p3t+0.4689p1t-τ1,p3˙t=0.305.3420-7.8599p3t+0.5342p1t-τ1+0.5876p2t-τ2.From (10) and (11), we can get ω20=3.612, τ20=0.508. In order to facilitate the calculation, let τ2=0.45∈0,τ20. On the basis of (18), we have τ10=0.547, S1S3+S2S4=512.74≠0, so (H1)-(H2) hold. From the conclusion of the third section, we know that the Nash equilibrium point E∗ is asymptotically stable when τ1∈0,τ10 and unstable when τ1>τ10. As τ1=τ10, Hopf bifurcation will occur.


### 4.1. The Influence ofτ1 on the Stability of System (22)

Figures1(a) and 2(a) show that system (22) is stable when τ1=0.530<τ10=0.547. When τ1=0.560>τ10=0.547, the system is unstable. This phenomenon can be found in Figures 1(b) and 2(b). The numerical simulation is consistent with the theoretical analysis.Figure 1
The time series of system (22) when τ2=0.45.
(a)
τ
1
=
0.530
<
τ
10
=
0.547
(b)
τ
1
=
0.560
>
τ
10
=
0.547Figure 2
The attractor of system (22) when τ2=0.45.
(a)
τ
1
=
0.530
<
τ
10
=
0.547
(b)
τ
1
=
0.560
>
τ
10
=
0.547Figure3 describes the process of system (22) from stable into chaos. From Figure 3(a), we can find that the system has bifurcation, and τ1 has the greatest impact on p1 and has less influence on p2. The change trend of the Lyapunov exponent in Figure 3(b) verifies the conclusion of Figure 3(a)  p3. We clearly find the bifurcation of system (22) when τ1=0.547 in Figure 3(b). Therefore, for enterprises in the price decision, it is necessary to ensure that τ1<0.547 when τ2=0.45.Figure 3
The influence ofτ1 on system (22) when τ2=0.45.
(a)
Price bifurcation
(b)
The biggest Lyapunov exponent


### 4.2. The Influence ofτ1 on Initial Value Sensitivity

If we take the initial value ofp1 is 0.4 and 0.401, respectively, the value of p1 will change after iterations. When τ1=0.530<τ10=0.547, after 61 iterations, the difference of p1 is 6.144 times of the initial difference 0.001. It can be described by Figure 4(a). In Figure 4(b), when τ1=0.560>τ10=0.547, after 61 iterations, the difference of p1 is 56.16 times of the initial difference 0.001. At this point, the value of p1 has strong dependence on the initial value. Therefore, we know that system (22) is already in chaos. So we can infer that price decisions-making will have many unpredictable, tiny price adjustments which will have a greater price deviation.Figure 4
The sensitivity ofp1 to initial value (0.4,0.401) when τ2=0.45.
(a)
τ
1
=
0.530
<
τ
10
=
0.547
(b)
τ
1
=
0.560
>
τ
10
=
0.547


### 4.3. The Influence ofτ1 and τ2 on Stability of Price

We takeτ1 and τ2 as parameters to study the effects of τ1 and τ2 on the price stability. With the increase of τ1 and τ2, the price changed from stable to unstable in Figure 5. When τ1 is greater than 0.52, the price will experience fluctuations; when τ2 is more than 0.5, the price will lose stability. When price is stable, the price will be stable at 0.8723. When price is chaotic, the highest price is 2.967 for τ1=0.8, τ2=0.15; the lowest price is 0.01871 for τ1=0.75, τ2=0.75. Therefore, enterprises should ensure that τ1 and τ2 are in a reasonable range when the price is set.Figure 5
The influence ofτ1 and τ2 on price.


### 4.4. The Influence ofτ1 and τ2 on Stability of Profit

We can see from Figure6, when τ1>0.52, the profit will be unstable; when τ2>0.5, the profit will fluctuate. As profit is in stable condition, the profit is stable at 1.367. When the profit is in an unstable state, the maximum profit is 1.367; the lowest profit is −13.6 for τ1=0.8, τ2=0.2. Through the analysis we can know that with the increase of τ1 and τ2, profit will decline but not higher than the stable value. Therefore, enterprises must maintain a reasonable value of τ1 and τ2; otherwise there will be a loss.Figure 6
The influence ofτ1 and τ2 on profit.


## 4.1. The Influence ofτ1 on the Stability of System (22)

Figures1(a) and 2(a) show that system (22) is stable when τ1=0.530<τ10=0.547. When τ1=0.560>τ10=0.547, the system is unstable. This phenomenon can be found in Figures 1(b) and 2(b). The numerical simulation is consistent with the theoretical analysis.Figure 1
The time series of system (22) when τ2=0.45.
(a)
τ
1
=
0.530
<
τ
10
=
0.547
(b)
τ
1
=
0.560
>
τ
10
=
0.547Figure 2
The attractor of system (22) when τ2=0.45.
(a)
τ
1
=
0.530
<
τ
10
=
0.547
(b)
τ
1
=
0.560
>
τ
10
=
0.547Figure3 describes the process of system (22) from stable into chaos. From Figure 3(a), we can find that the system has bifurcation, and τ1 has the greatest impact on p1 and has less influence on p2. The change trend of the Lyapunov exponent in Figure 3(b) verifies the conclusion of Figure 3(a)  p3. We clearly find the bifurcation of system (22) when τ1=0.547 in Figure 3(b). Therefore, for enterprises in the price decision, it is necessary to ensure that τ1<0.547 when τ2=0.45.Figure 3
The influence ofτ1 on system (22) when τ2=0.45.
(a)
Price bifurcation
(b)
The biggest Lyapunov exponent


## 4.2. The Influence ofτ1 on Initial Value Sensitivity

If we take the initial value ofp1 is 0.4 and 0.401, respectively, the value of p1 will change after iterations. When τ1=0.530<τ10=0.547, after 61 iterations, the difference of p1 is 6.144 times of the initial difference 0.001. It can be described by Figure 4(a). In Figure 4(b), when τ1=0.560>τ10=0.547, after 61 iterations, the difference of p1 is 56.16 times of the initial difference 0.001. At this point, the value of p1 has strong dependence on the initial value. Therefore, we know that system (22) is already in chaos. So we can infer that price decisions-making will have many unpredictable, tiny price adjustments which will have a greater price deviation.Figure 4
The sensitivity ofp1 to initial value (0.4,0.401) when τ2=0.45.
(a)
τ
1
=
0.530
<
τ
10
=
0.547
(b)
τ
1
=
0.560
>
τ
10
=
0.547


## 4.3. The Influence ofτ1 and τ2 on Stability of Price

We takeτ1 and τ2 as parameters to study the effects of τ1 and τ2 on the price stability. With the increase of τ1 and τ2, the price changed from stable to unstable in Figure 5. When τ1 is greater than 0.52, the price will experience fluctuations; when τ2 is more than 0.5, the price will lose stability. When price is stable, the price will be stable at 0.8723. When price is chaotic, the highest price is 2.967 for τ1=0.8, τ2=0.15; the lowest price is 0.01871 for τ1=0.75, τ2=0.75. Therefore, enterprises should ensure that τ1 and τ2 are in a reasonable range when the price is set.Figure 5
The influence ofτ1 and τ2 on price.


## 4.4. The Influence ofτ1 and τ2 on Stability of Profit

We can see from Figure6, when τ1>0.52, the profit will be unstable; when τ2>0.5, the profit will fluctuate. As profit is in stable condition, the profit is stable at 1.367. When the profit is in an unstable state, the maximum profit is 1.367; the lowest profit is −13.6 for τ1=0.8, τ2=0.2. Through the analysis we can know that with the increase of τ1 and τ2, profit will decline but not higher than the stable value. Therefore, enterprises must maintain a reasonable value of τ1 and τ2; otherwise there will be a loss.Figure 6
The influence ofτ1 and τ2 on profit.


## 5. Chaos Control

From the above analysis, we realize that the price and profit are in a state of chaos, which can lead to the fluctuation of the price and the profit. Therefore, we should take measures to prevent the system from entering a chaotic state or make it recover to a stable state. Below we take the method of the state variables feedback and parameter variation to control the system. Letτ1=0.58 and τ2=0.45; we can find that the system is chaotic from Figure 5. The time series and attractor of system (2) when τ1=0.58, τ2=0.45 are shown in Figure 7.Figure 7
The time series and attractorτ1=0.58, τ2=0.45.
(a)
Time series
(b)
AttractorAdding control variableμ in system (22), then system (22) becomes(23)p1˙t=1-μ0.205.0960-6.4614p1t-τ1+0.3058p2t-τ2+0.3567p3t+μp1t,p2˙t=1-μ0.255.2100-7.1470p2t-τ2+0.4168p3t+0.4689p1t-τ1+μp2t,p3˙t=1-μ0.305.3420-7.8599p3t+0.5342p1t-τ1+0.5876p2t-τ2+μp3t.The effect of μ on system (23) is shown in Figure 8. We can get that when μ=0.1055, system (23) has bifurcation phenomenon. That is to say, when μ<0.1055, system (23) is chaotic, and when μ>0.1055, system (23) is stable. With the increase of μ, the system changes from chaotic state to stable state.Figure 8
The influence ofμ on the system when τ1=0.58, τ2=0.45.
(a)
Bifurcation
(b)
Lyapunov exponentsLetμ=0.05<0.1055; we can find that system (23) is chaotic from Figure 8. The time series and attractor of system (23) are shown in Figure 9.Figure 9
The time series and attractor whenμ=0.05<0.1055.
(a)
Time series
(b)
AttractorLetμ=0.15>0.1055; we can see that system (23) is stable from Figure 8. The time series and attractor of system (23) are shown in Figure 10. Compared with Figures 9 and 10, chaos is controlled. The bigger the value of μ is, the more obvious the control effect is.Figure 10
The time series and attractor whenμ=0.15>0.1055.
(a)
Time series
(b)
Attractor


## 6. Conclusions

The model of [19] was improved considering three enterprises are bounded rationality and using the cost function under the resource constraints. At the same time, delay strategy was used by the first and second enterprises. Firstly, when τ2 is fixed, the influence of τ1 on the stability of the system is considered. Secondly, the effects of τ1, τ2 on the stability of price and profit were studied. The research shows that the value of τ1 and τ2 must be ensured in a reasonable range, and the price and profit are stable; otherwise there will be violent fluctuations. Finally, measures are taken to control chaos of system (2) successfully. The results of the paper play an important guiding value for the enterprise to carry on the price decision.



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*Source: 2717352-2017-04-10.xml*