一类二维非线性周期微分方程在退化平衡点的约化

On The Reducibility for A Class of Two-Dimensional Nonlinear Periodic Differential Equations with Degenerate Equilibrium Point

考虑一类二维周期微分方程在退化平衡点附近小扰动下的约化问题.通过引入外部参数,对系统进行仿线性周期变换,将问题化成具有外部参数的系统的约化问题,再由隐函数定理和拓扑度性质,得到原方程的一个约化的标准型.此外,在零点附近得到了一个周期解.

Consider the reducibility for a class of Two-Dimensional nonlinear periodic differential equations with degenerate equilibrium point under small perturbation. By introducing external parameters and making a shifting of the variables, ths equation is changed to one with external parameters. Furthermore, we get a normal form of the equation by the Implicit Function Theorem and the Topological Degree Theory. Moreover, we obtain a periodic solution near the origin.