一类时间可逆系统的逆积分因子

Inverse integrating factors for a class of time-reversible systems

由于含有系统的许多有用信息,逆积分因子被认为是解决常微分方程定性理论中两大公开问题:中心焦点问题、希尔伯特第十六问题的统一工具.而且逆积分因子与常微分方程的李对称性和Darboux可积性有密切的联系.因此对某些解析微分系统建立其逆积分因子的结构定理是重要的.对于一类时间可逆解析微分系统,建立了逆积分因子的系数递推公式.利用此递推公式得到其具有给定形式逆积分因子的充要条件.为了说明我们的结论,对于一个具体的时间可逆五次微分系统,利用系数递推公式直接给出系统的多项式型逆积分因子和初等首次积分.

Since inverse integrating factor contains a lot of useful information,it is deemed as a unified tool to deal with two main open problems： center-focus problem and Hilbert sixteenth problem in qualitative theory of ordinary differential equations.And it also has a close relationship with Lie symmetry and darboux integrability of ordinary differential equations.The refere,it is important to find a structure theorem of inverse integrating factor for some analytic differential systems.For a class of time-reversible analytic differential systems,coefficients recurrence formulae of inverse integrating factors are obtained,and the necessary and sufficient conditions for the systems to have prescribed inverse integrating factor are reached.To illustrate our results,polynomial inverse integrating factor and elementary first integral of a fifth reversible system are obtained directly by the coefficients recurrence formulae.