van der Pol-Duffing时滞系统的稳定性和Hopf分岔

研究了具有三次项的ｖａｎ ｄｅｒ Ｐｏｌ－Ｄｕｆｆｉｎｇ非线性时滞系统的稳定性和Ｈｏｐｆ分岔，分析了当线性化特征方程随两参数（时滞量和增益系数）变化时特征根的分布；证明了Ｈｏｐｆ分岔的存在性，通过构造中心流形并且使用范式方法给出的Ｈｏｐｆ分岔的方向以及周期解的稳定性，讨论时滞量对该系统的Ｈｏｐｆ分岔的影响。

STABILITY AND BIFURCATIONS IN A VAN DER POL-DUFFING TIME-DELAY SYSTEM

In this paperan, we consider a van der Po-Duffing time-delay system with cubic Nonlin-earity) givingwhere 0 < E < 1, A > 0, r > 0 is a delay, and first determine the Asymptotic stability of the'trivial equilibrium solution. A characteristic equation for the linearized system at the equilibriumis derived. The distribution of roots of the characteristic equation is analyzed, as a function oftwo parameters describing and A. The existence of Hop f bifurcation is proved when the systemloses the linearly asymptotic stability. The system is reduced to be a 2-dimensional equation byconstructing a center manifold. The direction of the Hopf bifurcation and the stability of theperiodic solution due to the Hop f bifurcation are diScussed by using the normal form. The resultsindicates that (i) the periodic solution of the system under consideration may be captured by asimple dynamical equation although the system is infinite dimension; (ii) the Hopf bifurcation issubcritical and the periodic solution is unstable; (iii) the time delay has a intrinsic effect on Hopfbifurcations and periodic solutions. Therefore, it is necessary further to investigate mechanism.