一类退化非线性微分方程的正规形计算

Computation of Normal Forms for a Class of Degenerate Nonlinear Differential Equations

对于退化非线性微分方程,给出了其主微分方程的保守-耗散分解,并证明了这种分解的几个性质。利用这些性质,把求定义在齐次向量场空间上的同调算子值域补空间,转化为求定义在齐次多项式空间上李导数算子值域补空间。在主微分方程是哈密尔顿的并且哈密尔顿函数在复多项式环C[x,y]上的因式仅为单因式的假设下,为求得系统的正规形,只需求有限个定义在齐次多项式空间上的李导数算子值域补空间,并给出递推公式。用该方法可求出一类具有广义Hopf奇点的正规形,并利用李三角形方法给出正规形与原微分方程系数之间的关系。

For degenerate nonlinear differential equations,the conservative-dissipative splitting is given,and some properties of this splitting are proved.By using these properties,a complementary subspace to the range of the homological operator defined on the homogeneous vector field space can be expressed in terms of a complementary subspace to the range of the Lie derivative operator defined on the homogeneous polynomial space.Under the hypotheses that the leading part of the degenerate nonlinear differential equations is Hamiltonian and the associated Hamiltonian function only has simple factors in its factorization on the complex polynomial ring C[x,y],to obtain the normal form,it needs only to compute a certain number of the complementary subspaces to the range of the Lie derivative operators defined on the homogeneous polynomial spaces,a recursive formulae of the computation for all the complementary subspaces are given.Finally,by using this method the normal form of a class of the generalized Hopf singularity is computed,relationship between the coefficients of the normal form and the origin equations is given by means of the Lie triangle method.