Qualitative Analysis And Calculated Index Number At Higher Order Singular Point

Give the differential equations:where X(x, y), Y(x,y) ∈ C0(D), field D R2 .If (1) is linear differential equations, it is quite clear on the path curve structure at the singular point nearby. If (1) is unlinear differential equations, the main way to study the qualitative structure at singularity in the neighbourhood is to discuss if there is path curve approaching the singular point and how many there is, and so on.Trajectories orientations are very complex at higher order singular point nearby. While path curve approachs the limit singular point (0,0) following t → + ∞ or t → -∞, the path curve must be along some a certain direction. How many path curve is there approaching the limit point (0,0) along such direction? Using the y = ux Briot-Bouquet transform, some conclusions are given in this paper.And several examples are analysed using these conclusions.The paper also analyses the y = ux transform on geometric meaning. And using this transform, higher order isolated singularity (0,0) in x,y plain is separated into several points on axis of u in x, u plain. In this paper it is proved that the index number at singular point (0,0) in x,y plain equals the sum of index number at separate points in x,u plain using theory of vector field votation degree.The singularity index number is a quantity of topological characteristic.lt isone of integer.Linear singular point index number calculation problem and higher order one index number calculation problem where its corresponding main equations are linear have been solved successfully. In addition,using, algebraic tool of Cauchy-index ,a set of formula is given respectivedly about index number calculation at critical singularity and higher order isolated singularity to the main equations. It is convenient for index number calculating especially for the critical singularity. And index number calculation problem at singular point in analytic vector field is also solved perfectly.