具有单向替代性的伯川德双寡头模型动力学研究

The Dynamics of Bertrand Duopoly Model with One-way Substitution

以垂直产品差异为前提,用数理建模构建了具有单向替代性的伯川德双寡头模型.采用动态经济学理论以及非线性动力学方法,分析双寡头模型的局部动力学行为.主要讨论了Nash均衡点的局部稳定性,研究发现系统在Nash均衡点处只能发生Flip分岔,并且应用中心流形定理和范式理论严格证明了系统发生的是超临界Flip分岔.除此之外,借助Matlab工具对模型的内部复杂性进行数值仿真,讨论了参数连续变化对模型分岔的影响以及初值敏感性.随着参数连续变化,系统进入混沌时会引起动荡,导致企业产生负面效应,因此采用状态反馈和参数变量控制法对系统施加控制使其形成新的长期稳定的分岔结构.

On the premise of vertical products differences,a dynamic model of Bertrand duopoly with one-way substitution is established by using mathematical modeling.The local dynamical behavior of duopoly model is analyzed by using dynamic economic theory and nonlinear dynamic method.The local stability of Nash equilibrium is discussed and it is found that only Flip bifurcation at Nash equilibrium occurs in the system.In addition,the central manifold theorem and the normal form theory are used to prove that the system only goes through supercritical Flip bifurcation.Besides,the Matlab tool is used to simulate the internal complexity of the model,both the effect of parameter changes on bifurcation and the sensibility of initial values within the system have been discussed.The system will lead to the economic turbulence when it enters into chaos with the changes of parameter,and some negative effects to the firms.In order to solve the above problems,a control has been implemented on the system by using the state feedback and the parameter variation control method to achieve a long term stable bifurcation structure.