^{1}

^{2}

^{1}

^{1}

^{2}

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

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. [

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 [

This paper is organized as follows: in Section

The triopoly dynamic game model is developed in [

Because price information is asymmetry, we consider three companies are bounded rationality based on model [

In a competitive market, the equilibrium points must be nonnegative. Considering generality, we assume that

We study the existence of Hopf bifurcation of the system at

At this point, the characteristic equation of system (

If

The impacts of delay on the stability of system (

The parameters of system (

Figures

The time series of system (

The attractor of system (

Figure

The influence of

Price bifurcation

The biggest Lyapunov exponent

If we take the initial value of

The sensitivity of

We take

The influence of

We can see from Figure

The influence of

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

The time series and attractor

Time series

Attractor

Adding control variable

The influence of

Bifurcation

Lyapunov exponents

Let

The time series and attractor when

Time series

Attractor

Let

The time series and attractor when

Time series

Attractor

The model of [

The authors declare no conflicts of interest.

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.