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With application of nonlinear theory, this paper makes study on the long-term competition in a recycling price game model by manufacturers and retailers. The paper makes analysis on the local stability of the Nash equilibrium point and gives the corresponding stable region. It has been found that the stability of the whole system would be significantly impacted by the following factors which include adjustment speed of the recycling price, the proportion of recycled products by channels, the sensitivity of consumers for the recycling price, and the price cross-elasticity between two channels. By means of the simulation technology, the complexity of the recycling price in the system in the long-term competition has been demonstrated. Owing to the change of parameters, bifurcation, chaos, and other phenomena would appear in the system. When the system is becoming chaotic, the profit of the whole system decreased. All these show that the operational efficiency for the whole system will be impaired by the chaos. Effective chaotic control of the system will be realized by the use of the parameter adaptation method.

The natural environmental pollution caused by industrial development for centuries has been spreading for decades. With increase of people’s environmental awareness, the reverse supply chain based on recycling and remanufacturing of recycled products has become more increasingly necessary.

Some scholars have conducted extensive research on relevant problems about close-loop supply chain: selection of recollection channels, selection of sales channels, pricing and coordination, net designing, and so on. There mainly exist two categories of research: the first one is product competition, that is, the competition between new products and remanufactured products for the same manufacturer; the second one is the competition of members in the supply chain, that is, the competition between the manufacturer and the remanufacturer for the same product. Savaskan et al. [

The economic system, with people being involved in, is a complex and open system with information feedback capabilities. For a nonlinear system, its subsystems, when they are being combined, will not have a simple linear additive effect on the whole system; they will be mutually coupled which makes the system produce a qualitative change and show the nature and characteristics that the original subsystem does not have. Research on the oligopoly game model and its complexity has attracted attention of scholars for many years. In an oligarch competition, the firms are either output or price setters; see the studies by Cournot [

For many economic management systems, chaos is detrimental to the system operation. Therefore, effective methods should be employed for suppressing chaos in accordance with the characteristics of the system itself. Many scholars proposed a variety of methods for chaos controlling, such as OGY method, controlling method for variable feedback, and controlling method for delayed feedback. At present, two methods are mainly employed for chaos control [

Based on the previous research, for the recycling price this paper established a game model consisting of only one manufacturer and one retailer. Under the circumstances of not fully obtaining market information for the manufacturers and retailers, this paper studied the influence of various parameters on the stability of the system by using the pricing strategy of limited rationality and evaluated the operational performance of the system by taking the system profit as the index.

The structure of the paper is as follows: in the second part, we constructed the corresponding model and made analysis on each parameter; in the third part, the local stability of the Nash equilibrium point has been analyzed; in the fourth part, the characteristics of the system in the long-term competition process would be discussed with the means of simulation technology, and relevant performance evaluation would be performed; Finally, in the fifth part, the conclusions of the paper would be achieved.

Nowadays, many manufacturers, in addition to their own recycling work, also require retailers to carry out recycling work in the process of sales. This is mainly because retailers are much closer to the market and consumers that it is more conducive for recycling process. On the other hand, only those large retailers have the advantage of recycling, and their recycling amounts of used products might be unmatched by those small-scaled and fragmented retailers. Therefore, these retailers involved in recycling could have great oligopoly superiority in a recycling market. For example, in the home appliance market, Gome and Suning are two traditional large retailers whose market shares in sales and recycling could be unmatched by other small home appliances. The structure of supply chain is as shown in Figure

The structure of the supply chain.

We assume that in reality the manufacturer and the retailer would recycle used products together. According to the Bertrand game model, the amount of recycled products is relevant to the recycling price; when there exists more than one recycling company, the amount of recycled products will also be related to the recycling channels and prices offered by other retailers. This could be shown by using a model as follows:

The model shows that when the retailer in one channel raises the price, the amounts of recycled products will increase. The profits of the manufacturer and the retailer can be indicated as

From formula (

From formula (

The game between the manufacturer and the retailer is a long-term and continuous process. Their relationship could not be adjusted to the optimal position by only game playing. The price competition between them is a long-term, repeated, and dynamic process with complexity. In reality, the market information obtained by the retailer is often incomplete for the reason that obtaining more market information obviously requires more costs. In many cases, owing to the distorted and asymmetrical information, perfect information would be rare even if great efforts have been put into the market investigation. We assume that both the manufacturer and the retailer will adopt expectations of bounded rationality. This indicates that the retailer’s price decision in the next period relies on a local estimate of the marginal utility for the current period. If the marginal utility is positive in this period, the retailer will raise the price of the next period; if the marginal utility is negative, the retailer will lower the price of the next period. The dynamic adjustment process is as follows:

According to the definition of “fixed point,” in system (

According to the value range of the parameters, we know that these four equilibrium points are nonnegative and they have their particularly economic significance. Among them,

Jacobian matrix for system (

Similarly, the equilibrium point

Among them,

Its corresponding characteristic equation is

We got this here:

According to Jury’s condition for equilibrium point stability, we obtain the necessary and sufficient conditions for the local stability of the Nash equilibrium point

According to judgment condition for stability of the Nash equilibrium point, in the case of any two conditions being satisfied, the violation of the other one will cause complex dynamic behaviours for the system.

We put

At this time, condition (

From the perspective of our actual competition, we could give economic significance to the stability of the four equilibrium points in the market.

We try to know the relationship between various factors in the model by means of numerical simulation for better understanding of the dynamic characteristics in market competition. On the basis of actual competitive situation, we hold a view that it is more convenient for the retailer to recycle products at lower costs than the manufacturer.

According to the judgment condition (

Local stable region of the Nash equilibrium point about

Figure

Attraction domain of the system.

By the means of 2-D Bifurcation diagram we make analysis on the types of system bifurcation, the periodic behavior of the solution, and the path leading to chaos. Firstly, we use the adjustment factor of price input as the bifurcation parameter. Figure

The 2-D bifurcation diagram of the system with

Figure

Impact of consumer sensitivity on recycling prices on system stability.

Impact of price cross-elasticity between two channels on system stability.

For the purpose of further analysis on the dynamics of the system, the influences of parameters on attractors’ basin of attraction will be studied through the critical curves. LC (critical curves) is defined as the locus of points which have two (or even more) coincident rank-1 preimages. These preimages are located on a set and are called LC_{−1}. LC can subdivide the plane into different regions in accordance with the number of their preimage (see [Puu,1996] for more details); LC_{−1} belongs to the points where the value of Jacobian determinant is 0.

System (

Figure _{1} when red lines are LC_{−1} and black lines are LC (the same below). From Figure

The basin of attraction when

The basin of attraction when

For better understanding of market competition behaviors after the system has entered into chaos, we will make analysis on the complexity of the long-term game for system (

Figure

Bifurcation diagram and Lyapunov exponents with

Singular attractor, the result of interaction between the overall stability and local instability of the system, is another feature of system chaos. It has self-similarity and a fractal structure. Figure

Attractors of the system get changes with the increase of

When the system is in the chaotic state, any changes of the initial value will have a greater impact on the result, even if these changes are very tiny and small. That is to say, the evolution result of the system has an extremely sensitive dependence on the initial value, which is the so-called Butterfly Effect. The red lines area in Figure

Power spectrum of the system when

Sensitivity dependence to initial condition when

Figure

Profits when the system is at different times.

Different periods | Stable period | Two-period-doubling bifurcation | Four-period-doubling bifurcation | Chaotic period |
---|---|---|---|---|

| 7.4796 | 7.2040 | 6.5778 | 6.1118 |

| ||||

| 12.7450 | 12.9026 | 13.2846 | 13.6014 |

| ||||

| 20.2247 | 20.1066 | 19.8624 | 19.7132 |

Changes for profits in different periods: (a) steady state; (b) chaotic state.

From the above analysis, we can learn that, in the complex price competition, the market equilibrium is only a short-term process in which many changeable factors are involved. For example, when the retailer speeds up price adjustments, the market will change from the stable state to chaotic one. Once the market enters into chaos, neither the manufacturer nor the retailer could predict the market competition, the pricing will become disordered, and the market will not achieve an effective equilibrium. The above analysis also shows that the retailer would get lower profits in this state than in the balanced state. In the actual market competition, retailers will adopt various competition strategies for profit maximization after long-term price games. Price competition is their favorite and most frequently used competition method, while excessive price competition would easily lead to inefficiency and disorder for the market and bring losses to retailers. Therefore, it is very important to take timely control measures for the market’s returning to a stable equilibrium.

The method of parameter adaptation is employed to control the system effectively (see the study by Huang [

In the control system (

Bifurcation diagram with respect to diffident K: (a) K=0.65; (b) K=0.7.

K=0.65

K=0.7

Bifurcation diagram with change of the control parameter K.

Power spectrum for price: (a) K=0; (b) K=0.3.

In this paper, for the study of the long-term competitive behaviour of the manufacturer and retailer in the process of recycling products, we established one recycling price game model which consists of one manufacturer and one retailer. Analysis was made on the stability of the equilibrium point in the game model with the result that three unstable bounded equilibrium points and a Nash equilibrium point with local stability have been founded. The stability conditions of the Nash equilibrium point have got deeper research. The complexity behaviour of the system is studied by using bifurcation diagram, maximum Lyapunov exponent diagram, chaotic attractor, and power spectrum. Finally, we make effective control of the chaotic behaviours in the system by the application of the control method for variable feedback. Through the research we can get the following conclusions:

No data were used to support this study.

The authors declare that they have no conflicts of interest.

The research was supported by China Postdoctoral Fund (No. 2016M602155), Ministry of Education Humanities and Social Sciences Project (No.18YJCZH081), Scientific Research Projects in Shandong Universities (No. J18RA055), Talent Introduction Project of Dezhou University (No. 2015skrc05), and The Subject of Bidding for Key Disciplines in the 13th Five-Year Plan of Dezhou University (No. 3010040235).