平均法与Hopf分叉的规范型系数

The Extended Averaging Methods and the Coefficients of Hopf Bifurcation's Normal Form

证明了非线性振动系统对应的平均方程是原系统的一个 Arnold-范式 ,并将平均法推广到了参数化系统 ,大大简化了 Hopf分叉范式及其系数的计算过程 ,同时 ,证明了在非退化情况下 ,ε3以上的项不影响 Hopf分叉三阶截断范式的系数。

This paper shows that the averaged equation of a nonlinear oscillation system is an Arnolodnormal form of the original system.In light of the averaging methods extended to the parameterized systems,3 order truncated normal form of Hopf bifurcation and its coefficients are calculated.The methods have shorten the calculation of normal form greatly and get the same results as the exact ones as well.It is shown that the terms of higher than ε 3 do not affect the coefficients of the 3-order truncated normal form of Hopf bifurcation.