一类时滞非线性振荡器的Zero-Hopf分支规范型计算

NORMAL FORM COMPUTION OF ZERO-HOPF BIFURCATION IN A CLASS OF DELAYED NONLINEAR OSCILLATORS

时滞反馈项在非线性振荡器中有着很重要的影响.为了深入研究时滞项在系统向量场中的作用,首先利用中心流形化简和规范型方法计算当系统向量场特征方程存在单零根和一对纯虚根、其他根有负实部时产生的Zero-Hopf分支并进行了分析研究;利用具体模型的数值模拟,阐明一类时滞非线性振荡器在时滞项变化时可以产生不同的类型的分支现象.对进一步研究不同系统模型产生何种分支现象有着重要的意义.

The effects of delayed feedback terms on nonlinear oscillators are very important. In order to study the effect of delayed terms on system vector fields,firstly,using center manifold reduction and normal form method, we compute and analyze that the system appears Zero-Hopf bifurcation when its characteristic equation has a simple zero root and a pair of purely imaginary roots and all other roots have negative real parts. Then,we clarity that a class of delayed nonlinear oscillators show different bifurcation when the delayed terms change via numerical simulation of specific examples. It’s significant to investigate the other different systems further and what types of bifurcations may occur on systems.