| As we know,studying the variation law of objects in the objective world can be summarized as studying qualitative characteristics of solutions to the differential equation.After the collaborative efforts of Chinese and foreign experts and scholars,abundant fruits have been achieved for autonomous differential equation,especially for the polynomial differential system.However,there are few achievements on the non-autonomous differential systems due to complexity of models and rarity of good methods.In 1980s,Mironenko,a differential equation expert from the Former Soviet Union,proposed the concept of reflection function for the first time and constructed a reflection function theory.This provides a new way for people to explore geometric characteristics of solutions to the complicated differential system.Subsequently,people began to study qualitative characteristics of micro-solutions to the non-autonomous differential system based on this theory and achieved outstanding results.It can be seen from the reflection function theory that if the reflection function of a differential system is known,it can be used to study geometric characteristics of its solution.In particular,the Poincare mapping of the periodic differential system can be constructed by the reflection function,thus enabling to determine geometric characteristics of its periodic solution.The key to solve these problems is how to calculate expression of reflection function of the differential system or structural forms and properties of the reflection function.In this study,the structural form of reflection function of the generalized n-order polynomial differential system was studied and it was used to discuss geometric characteristics of its solutions.(?)(1)Key research contents are introduced as follows.The structural form of the second component of the reflection function in the differential system(1)was explored when the first component doesn’t depend on y.Meanwhile,sufficient conditions of reflection functions which have such forms in the differential system(1)were proposed.Moreover,parity and symmetry of solution of the differential system(1)as well as geometric characteristics of its periodic solution and characteristics of equivalent differential system were discussed by the conclusions. |
