Perturbation Analysis Of A Central Focus Of The System

In the research of qualitative theory of ordinary differential equations, the judge of center-focus piont is a kind of important issues.The research of center-focus piont will ultimately depends on the calculation of focus piont, so that is one of the most basic content in the research. Subsequent function method and form progession method are two classical methods in the caculation of focus point. In this paper, the author give a quartic homogeneous system,caculate its saddle point by use of the subsequent function method.In order to get a system's saddle point.They can solve the issue of the calculation of saddle by using the relationship between focus point and saddle point,also,we calculated the formula of subsequent function by using a new method of curvilinear coordinates. the author hope that some essential characteristics can be found through the curvilinear coordinates system and decrease the calculation at the same time,they also expect to offer some help to solve the problem of the bound of three systems's focus.The whole paper are divided into four parts.The first chapter is an introduction,we mainly introduced the history of the development of quality theory of differential equation,the main content in qualitative theory research, and some others'main research methods and results of calculating the focus in recent years, put forward to the relationship between the saddle point and focus point. At last we simply described the main work of this aricle.In the second chapter,the author give a general system of degree four with weak critical singularity and calculated the saddle point. For an arbitrary system, They need to calculate the focus,due to the calculation of saddle points is relatively simple than the calculation of focus,so they can translate the original known system into the new system which has the fine saddle point through a transformation,then calculate the new system's saddle points by use of the subsequent function method.They can get the original system's focus through the relationship between the parameters of the two systems. At last,we made a concrete example, calculated the first three order saddle point.In the third chapter, the author translated a general system of degree four with weak critical singularity into the general curvilinear coordinates system by curvilinear coordinates, put forward the relationship between curvilinear coordinates and right-angle coordinate,and calculated the formula of subsequent function with curvilinear coordinates, So they can show the focus formula of a system with degreen n, avoiding the using of unnecessary operation which will brought by polar coordinate Finally, we made a summary to the whole paper and an outlook about the research of the central focus in the qualitative theory of ordinary differential equations.We hope that based on this paper, there could have a breakthrough in the future works.