Chaotic Dynamic Of Bowley Model

There are many articles to study duopoly game, but they base on the totally rational foundation basically, most articles focus on to discuss the static equilibrium. Especially they mainly discuss equilibrium solution of duopoly games, seldom solving non-equilibrium solution and chaos dynamic analysis. The non-balanced theory think that under different economic systems it is different to deviate from the balanced dynamic course, therefore it is different to realize balanced interaction, this makes non- balanced theory focus on economical operation reality, probe into various kinds of expression forms of balanced orbit existing actually and the way to realize it.In view of the above, this article apply the nonlinear science to duopoly game, such as chaos, fractal, microeconomics, industry's economics,etc, by means of contrast, reasoning and studied numeric simulation,etc, We study the chaotic dynamics and the parameter controls of Bowley model under the parameter condition and the directive significance to enterprise's market behavior.Firstly we review the current duopoly model via game theory, firms usually adopt simpler rule that is a decision-making course. We deduce that heterogeneous expectations may lead to a rich dynamics and complexity. Some types of players are considered: naive, bounded rational and adaptive expectations. In this study we show that the dynamics of the duopoly game with players, whose beliefs are heterogeneous, may become complicated. Then we analyze the dynamics of Bowley model with delayed bounded rationality, we show that firms using delayed bounded rationality have a higher chance of reaching Nash equilibrium and numerical simulations are used to show bifurcations diagrams. We analyze a Bowley duopoly game with spillover effect, where players have heterogeneous expectations. And we also show that the firm with spillover effect has a higher chance of reaching Nash equilibrium. As other parameters of the model are fixed, the greater the difference of product is, the less the coordinates the Nash equilibrium point is, the bigger of the stability region of the Nash equilibrium point is, then the complex (periodic or chaotic) behavior occurs. A Bertrand model with bounded rationality is studied. We analyze the existence and stability of the nonlinear system, and observe the complicated phenomenon such as bifurcation and chaos. Numerical simulations are presented to show that players with heterogeneous beliefs make the duopoly game behave chaotically. We also study the influence and control of chaos and gain some helpful conclusions. Lastly we also evaluate system performance, and we control the chaos with different methods. We also discuss the result of the regulating behavior of the firm to realize the equilibrium state. These are illuminated by numerical simulation.