The Research Of Singularity And Limit Cycle For Several Types Of The Differential System

For differential equation, it solves some practical problems about human’s activity. In daily life, a lot of problems can be converted into differential equation problem through research and analysis, such as ecosystems, cybernetics, electronic device design, the qualitative analysis of ballistic and the flight path, chemical analysis, population model, etc. It has a lot of influence in astronomy, physics, aerospace and other fields of science. The hundred years passed, after the exploration and research of many famous mathematicians like poincare’ and LiYapurove all over the world, the theory of differential equation gradually improves. Among them, the theory of differential system ring domain is one of the most practical theory and has a very important role in research about the singularity and limit cycle of the plane autonomous system.Since Hilbert raise the problem of the second part of 16th on international conference, the number and position of the limit cycle for plane system gets widely attention.As for a large number of studies of Lie’nard Lie system which is similar to Lie’nard equation, many researchers use the real domain qualitative theory of ordinary differential equation and get the existence of limit cycle with sufficiency theorems under different conditions. And the theorem of the plane autonomous system plays an unimaginable work in the process of the research. Many complex differential system can be transformed into Lie’nard system though a series of topology, and based on the adequacy of related theorem analysis. For example, according to different unknown parameters, the limit cycle of quadratic and three of plane system get the comparatively detailed research,, a lot of literature [1-7] got some very good conclusions; for the higher order perturbation system is got by special Hamilton system.the limit cycle of these particular systems is given the number of the biggest upper bound in literature [8-16]. Due to the higher number of plane system, the unknown parameters, the more the number of the limit cycle of high order system upper bound is not known.As is known to all, the singularity and the limit cycle are closely linked, the interior of the limit cycle contains the singularity, and the sum of the singularity index is 1.Therefore, we must study the properties of the singularity before the research of location of the nontrivial periodic solution. The judgment of the existence of nontrivial periodic solutions area roughly according to the sum of the singularity index, it can avoid many unnecessary assumptions and greatly reduce the relevant workload.This article firstly introduces research background of differential equations and summarizes the research works. Secondly, we use a technique to calculate higher order singular index of a two-dimensional system and give some new conclusions by translating the singularity index into the zero solution of equations. Thirdly, we study limit cycle of a class of E31 system which contains three singularity The sufficient conditions for existence and uniqueness of limit cycle is given by using the contradiction between the singularity index and the limit cycle index and Hopf bifurcation theory and of Lie’nard system theory. Finally, we study the number and position of limit cycles of a kind of a En+11 system, and obtain the upper of number and. position of limit cycles by transforming the problem-for the number of limit cycles into the zero root of polynomial.