| Ordinary differential equation qualitative theory has become an indispensable tool in the celestial mechanics, space technology, satellite communications, and other areas of the cutting-edge research. Moreover, it has been widely applied in biological, medical, modern chemistry, physics and other fields. However, in the qualitative theory research, the existence and uniqueness of solutions, and the research of limit cycles occupy an im-portant position. In this paper, we mainly study the general two-dimensional autonomous system, a class of nonlinear system in biochemical reactions and its high order system, specific work summarized as follows:Chapter One, we firstly introduce the research background and practical significance, and then give the main research work in this paper.In the second chapter, we mainly study the generally two-dimensional autonomous system by Osgood condition and the implicit function existence theorem, and obtain the weak and simple condition of uniqueness theorem of solutions of the system, etc. In addition, a specific example is given to illustrate the feasibility of the conclusion.In the third chapter, we mainly discuss a class of nonlinear system in biochemical re-action by introducing the Dulac function, combining with Poincare-Bendixson ring domain theorem, and using Differential equation of geometric theory, then obtain the existence and nonexistence of its limit cycle. In addition, we also get the further conclusion than former by using the LINGO software.In the fourth chapter, based on the existing research results in the previous chapter, we study the high order nonlinear system in biochemical reactions of the previous chapter, and gain the conclusion of its limited singularity and point at infinity, etc.In the fifth chapter, we summarize the work of the full text, and the research prospect of the high order nonlinear system in biochemical reactions is discussed. |
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