## Theoretical and Computational Advances in Nonlinear Dynamical Systems

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

**Academic Editor:**Kaliyaperumal Nakkeeran

#### 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 parameters and , respectively. According to the change of delay parameters and , 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 and . 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.

#### 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 is overcome, , and , 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 Section 2, 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:where , , represents the largest market demand for products, is elastic demand, , represents the substitution rate between the two companies, respectively, , denote the price and output of the product, respectively, represents the weight of the current price, and represents the weight of price of time. The cost function with linear form is , , and is marginal profit. In (1), the first enterprise adopts the adaptive pricing strategy with delay, where denotes the delay parameter; the other two enterprises employ the finite rational pricing strategy. In addition, the second enterprise used the postponement strategy, where stands for the delay parameter. The linear cost function under the condition of sufficient resources was used. Then 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 and , respectively. The cost function will obviously increase under limited resources; that is, , where is the fixed cost. We further have the improved model with price game:

#### 3. Local Stability at Equilibrium Points

In a competitive market, the equilibrium points must be nonnegative. Considering generality, we assume that is a Nash equilibrium point of model (2), where

We study the existence of Hopf bifurcation of the system at . Let , , and , with , , and instead of , , and , respectively, when . We have the linear form of the system through Jacobian matrix as follows:The determinant of (4) iswhereTherefore the characteristic equation for system (4) iswhereWe discuss the effects of , on the stability of system (4) when , , .

At this point, the characteristic equation of system (4) isWe consider (2) with in its stable range, regarding as a parameter. Taking into account the generality, we discuss system (2) under the case mentioned in [14], and . is defined as in [14]. Therefore we have where , , are the positive roots ofwhereLet be a root of (9). Then whereFor (13), we can obtain thatFor (15), we obtain the following equation:Suppose that : (16) has finite positive roots. We define the roots of (16) as . For every fixed (), there exists a sequence which satisfies (16). It isLetWhen , (9) has a pair of purely imaginary roots for . In the following, we take the derivative of with respect to in (9) for the transversality condition of Hopf bifurcation, and we haveThen we further havewhereObviously, if , based on the above discussions and by the general Hopf bifurcation theorem in [15], we can obtain the results as follows.

If hold, when , then the Nash equilibrium point of system (2) is asymptotically stable for and it is unstable as . System (2) will be under Hopf bifurcation at when .

#### 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 , , , , , , , , , and ; the marginal costs of three dairy product companies are , , and ; the initial prices of their products are , , and ; the speeds of price adjustment are ; the fixed cost of the enterprise is , , and . Considering the following system, it is easy to calculate the Nash equilibrium point of system (2) which is .From (10) and (11), we can get , . In order to facilitate the calculation, let . On the basis of (18), we have , , so hold. From the conclusion of the third section, we know that the Nash equilibrium point is asymptotically stable when and unstable when . As , Hopf bifurcation will occur.

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

Figures 1(a) and 2(a) show that system (22) is stable when . When , 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.

**(a)**

**(b)**

**(a)**

**(b)**

Figure 3 describes the process of system (22) from stable into chaos. From Figure 3(a), we can find that the system has bifurcation, and has the greatest impact on and has less influence on . The change trend of the Lyapunov exponent in Figure 3(b) verifies the conclusion of Figure 3(a) . We clearly find the bifurcation of system (22) when in Figure 3(b). Therefore, for enterprises in the price decision, it is necessary to ensure that when .

**(a) Price bifurcation**

**(b) The biggest Lyapunov exponent**

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

If we take the initial value of is 0.4 and 0.401, respectively, the value of will change after iterations. When , after 61 iterations, the difference of is 6.144 times of the initial difference 0.001. It can be described by Figure 4(a). In Figure 4(b), when , after 61 iterations, the difference of is 56.16 times of the initial difference 0.001. At this point, the value of 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.

**(a)**

**(b)**

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

We take and as parameters to study the effects of and on the price stability. With the increase of and , the price changed from stable to unstable in Figure 5. When is greater than 0.52, the price will experience fluctuations; when 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 , ; the lowest price is 0.01871 for , . Therefore, enterprises should ensure that and are in a reasonable range when the price is set.

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

We can see from Figure 6, when , the profit will be unstable; when , 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 , . Through the analysis we can know that with the increase of and , profit will decline but not higher than the stable value. Therefore, enterprises must maintain a reasonable value of and ; otherwise there will be a loss.

#### 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. Let and ; we can find that the system is chaotic from Figure 5. The time series and attractor of system (2) when , are shown in Figure 7.

**(a) Time series**

**(b) Attractor**

Adding control variable in system (22), then system (22) becomesThe effect of on system (23) is shown in Figure 8. We can get that when , system (23) has bifurcation phenomenon. That is to say, when , system (23) is chaotic, and when , system (23) is stable. With the increase of , the system changes from chaotic state to stable state.

**(a) Bifurcation**

**(b) Lyapunov exponents**

Let ; we can find that system (23) is chaotic from Figure 8. The time series and attractor of system (23) are shown in Figure 9.

**(a) Time series**

**(b) Attractor**

Let ; 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 is, the more obvious the control effect is.

**(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 is fixed, the influence of on the stability of the system is considered. Secondly, the effects of , on the stability of price and profit were studied. The research shows that the value of and 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.

#### Conflicts of Interest

The authors declare no conflicts of interest.

#### Acknowledgments

The paper is supported by “The Fundamental Research Funds for the Central Universities,” South-Central University for Nationalities (CSY13011). The authors extend their gratitude to Fengshan Si, Yuhua Xu, and Junjie Li for their help in model building and computing.

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Copyright © 2017 Yuling Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.