非线性Mathieu方程的全局分叉

Global Bifurcation of Mathieu Nonlinear Equation

将多尺度Ｌ－Ｓ方法用于研究参数激励振动系统的全局分叉，确定了该系统在主参数共振情况下的分叉集合，得到了某些余维二退化分叉点附近产生混沌运动的条件。

Mathieu equation represents a class of important systems in nonlinear vibration;By theintroduction of multiple scales,Liapunov-Schmidt method is reformed and used to obtain theaveraging system of Mathieu equation. By using normal form and universal unfolding theo- ries,the degenerate averaging systems are transformed into universal systems that can cap-ture the complete dynamic behavior. The stabilities of the singular points,the heteroclinicand homoclinic orbits of the averaging system are analyzed.By referring to the global bifur-cation results of the averaging system, it has been found that periodical vibration , ampli-tude-modulated vibration and chaotic motion exist in the nonlinear Mathieu vibration sys- tem,