Qualitative classification of singular points

We study in this paper the qualitative classification of isolated singular points of analytic differential equations in the plane. Two singular points are said to be qualitatively equivalent if they are topologically equivalent and furthermore, two orbits start or end in the same direction at one singular point if and only if the equivalent two orbits start or end in the same direction at the other singular point. The degree of the leading terms in the Taylor expansion of a differential equation at a singular point will be called the degree of this singular point. The qualitative equivalence divides the set of singular points of degree m into equivalence classes. The main problems studied here are the characterization of all qualitative equivalence classes and then, to determine to which class a singular point will belong to. We remark that up to now these problems have been solved only in the case m = 1 before. The difficulty for this classification problem is that the number of blowingups necessary for the analysis of a singular point is unbounded (although it is finite) when this singular point varies in the set of singular points of degree m. To overcome this difficulty, we associate an oriented tree to the blowing-up process of any singular point such that each vertex represents some singular point. Then we prove that: (i) the above unboundedness comes exactly from the arbitrary length of an equidegree path; and (ii) the local phase portraits of the starting and the ending singular points in such a path are closely related by a simple rule which depends on the parity of the length of the path, but not on the length itself. Thus we obtain a successful method for this classification which can be applied in principle to the general m-degree case. As application of our method, we get the precise list of qualitative equivalence classes of singular points of degree 2 (Theorem D). Further, we prove (Theorem F) that there are finitely many qualitative equivalence classes in the set of singular points of degree m, and there are a finite set of quantities, which are computed by a bounded number of operations, such that they determine to which class a given singular point belongs. As a by-product we obtain the topological classification of singular points of degree 2. A topological equivalence class is determined by the number of elliptic, hyperbolic and parabolic sectors (denoted by e, h and p respectively) of the local phase portraits and their arrangement. The precise conditions for the tripe (e, h, p) have been given by Sagalovich. But this result is not sufficient for the topological classification problem. In fact, according to Sagalovich's theorem, there are 15 possible topological equivalence classes in the set of all singular points of degree 2; our result (Corollary E) shows that there are exactly 14 classes.