Reflective Function Of The Cubic Polynomial Differential Systems

Eighties of the last century,the former Soviet Union experts in differential equations Mironenko create the theory of reflective function which provided a new avenues to study the state of the differential equations x ' = X(t,x).From now on,more and more people began to study the reflective function and the solution behavior of periodic systems with the help of reflective function.Many good results[20 ? 30]have been got when system is polynomial differential system special is two-degree polynomial differential system.On the basis of work which has been done,in the present paper we further research the reflection function of the three-degree polynomial differential system.It introduces the background, present status and significance of this article in introduction.Then,to convenient,it gives the detailed definition and the basic properties of the reflection function.We give a description that the theorem of the reflective function and the Poincarémapping.In the section,We introduce the basis concepts which will be used throughout the text from beginning to end.As a main part of the article,for the three-degree polynomial differential system Firstly,we studied the first component F1 of reflection function ( )F = F1 ,F2,F3T nothing to do with y ,z (i.e. ( F1 =α(t ) x)), throughing discussion, to prove that F2 ,F3 are both linear expression of y ,z .Secondly,we follow from the particular to the general treatment strategy,first considering when f 32(t , x )is odd (or even) function with t ,discussing the sufficient conditions of the differential system (1) with generalized linear reflection function then considering when f 32(t , x )is general case,the sufficient conditions of the differential system (1) with generalized linear reflection function (2).And when the given system (1) is 2ω- periodic system,the criterion of existence of periodic solution,as well as qualitative behavior of the solution.Finally,two examples are given in the fourth part of the article.In this paper,we extend the related conclusions of [28],[30] on the polynomial differential systems.