余维2齐次扰动系统的分支

讨论了一类在分支值线性部分具有两个零特征根且只有一个Jordan块,而扰动项为n次的齐次平面向量场.讨论此类系统的分支的一个重要工具是:Melnikov函数,然而当n较大时,不易得到相应的性质.引入了一类判断函数,通过对该判断函数性质的研究,基本上确定了该向量场的轨线分支图.

Bifurcations to codimension two system under homogeneous perturbation

In this paper,a kind of planar vector fields,whose linear part has two zero eigenvalues with one Jordan block and the nonlinear item is of power n,are considered.An important tool of discussing this system is the Melnikov function, however,it is not easy to get the properties when n is larger.A kind of judgement functions are introduced,and by studying the character of the judgment functions, the orbit bifurcation graphs of this vector field can be basically determined.