廖琴， 赵维锐^*

（  武汉理工大学 理学院， 武汉 430070；  ）

摘要： 主要考虑具有下临界的企业竞争模型，针对企业生产所产生的生产周期和信息时滞对企业生产产量的影响，将时滞引入具有下临界企业竞争模型中，提出了具有时滞和下临界的企业竞争模型。以时滞为参数，在适当的条件下，得到了系统在内部平衡点处的局部稳定性和产生Hopf分支的条件，发现当参数经过临界值时，企业的生产产量产生周期性震荡。进一步利用中心流形定理和规范型方法，得到分支的方向和周期解的稳定性。最后通过数值模拟验证所得到的结论。
关键词： 下临界；稳定性；双时滞；Hopf分支；周期解

LIAO Qin， ZHAO Weirui^*

（  School of Science, Wuhan University of Technology, Wuhan 430070；  ）

Abstract： This paper mainly considers the enterprise competition model with lower critical, In view of the enterprise production cycle and information delay influence on enterprises production, The time delays is introduced into the competitive model of the lower critical enterprise, the enterprise competition model with time delays and lower critical is put forward .Based on time delay parameter, under appropriate conditions, the local stability of the system at the internal equilibrium point and the conditions of generating the Hopf bifurcation are obtained,It shows that when the parameter through the critical value, the production yield produce periodic shocks. Further use of the center manifold theorem and normal form method, the stability of the direction of bifurcation and periodic solution are obtained. Finally,the obtained results are verified by numerical simulation. 
Keywords： lower critical; stability; two delays;Hopf bifurcation; periodic solution

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