| In1981, the former Soviet Union expert in the differential equations Mironenko firstly created the theory of reflection function. It is a new tool to find the system Poincare mapping by means of reflection function, thereby inaugurated a new field of differential equation. In recent years, more and more experts and scholars has made many good results in this direction. It is a new task to use of reflective function theory to study the behavior of solutions of periodic system. The founder Mironenko proposed some new problems, today we studied one of them, and drawn some conclusions, which we further explain the laws of movement of objects and provide the new theoretical basis and new criterion.On the basis of work which has been done in the existing literature, we further research reflective function of the two-degree polynomial differential system. In the preface, it introduces the background, present status and significance of this introduction. Then, it gives the definition and the basic properties of reflection function and Poincare mapping. We introduce the preparatory knowledge which will be used throughout the text from beginning to end.This article follows methods which is from the particular to the general, the first part of the two-degree polynomial differential systemFirstly, we studied the first component F1of reflection function F=(F1, F2)T nothing to do with y, and got the form of the second component F2, ie. it is linear function of y. Secondly, we studied the first component F1of reflection function of y, and got the form of the second component F2. Finally, we discusses when the first component F1of reflection function was quadratic polynomial of x,y, the second component is also in the form of two-degrees polynomial.The second part discussed the sufficient conditions of the differential system with these forms, and when the system (1) is a2w-periodic system, we got the expression of Poincare mapping, and geometric properties of periodic solution. |
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