A Study On The Qualitative Relation Of Solutions Of Higher Dimensional Systems And Differential Equations

In the objective world the motion equation of many objects can be described by the differential system x’=X(t,x). While in the process of studying differential system, a central issue is how to determine the existence of the system’s periodic solutions and its qualitative behavior if it has. Generally, if X(t+2ω)=X(t,x)(ω is a positive constant), to study the solutions’ qualitative behavior, we could use the Poincare mapping,but sometimes it is very difficult to find the Poincare mapping for many systems. In the1980’s, the Russian mathematician Mironenko established the theory of reflective function. In recent years, more and more experts and scholars applied this theory to study the geometric properties of the differential system’s solutions, which provide us a new theoretical basis and criterion to further explain the laws of movement of objects.On the basis of the existing literature,we used the reflective function theory to establish a relationship between system (3.4) and an Abel or a Bernoulli equation or other one-dimensional differential equation. And then we study the qualitative behavior of the cubic polynomial differential system. In the introduction of this article,we have introduced the significance and status of our research. To facilitate the description, we gave the definition and the basic properties of reflective function and the relationship between reflective function and Poincare" mapping in preparatory knowledge, which will be used throughout the rest of this article.In this paper, we mainly studied the qualitative behavior of the differential system where a,(t), bi (t)(i=1,2,…,9) are continuously differentiable in R, and there exist a unique solution for the initial value problem.For system (3.4), first we studied the structure of F(t,x) if it satisfies the equations (3.2),(3.3) and got that which will either has the form of And then we discuss the sufficient conditions if system (3.4) has such structure forms of F(t,x). Meanwhile we got the qualitative relationship between the system (3.4) and an Abel or a Bernoulli equation or other one-dimensional differential equation respectively. At last we can draw a conclusion that to get some qualitative behavior of a cubic polynomial differential system’s solutions, we could just study the qualitative behavior of the solutions of its corresponding Abel or Bernoulli or other one-dimensional differential equation. Finally, we gave examples to verify the correctness and feasibility of the conclusions we got in this article.