Elementary Singular Point Research Articles

Note on a Theorem of Berlinskii

If a quadratic differential system has four singular points, these are elementary and the sum of their indices is 0 iff the quadrilateral with vertices at the singular points is convex; otherwise the sum of indices is 2 or -2. These facts and the relative positions of the two kinds of singular points are readily proved by consideration of the pencil of isoclines of the system. The theorem is originally due to Berlinskii. We refer to [1]. In that paper Coppel presents a theorem of Berlinskii with a proof by Kukles and Casanova. We give a simpler proof of purely geometrical character. THEOREM. Suppose that a real quadratic differential system has four critical points in the affine plane. If the quadrilateral with vertices at these points is convex then two opposite critical points are saddles and the other two are antisaddles (nodes, foci or centers). But if the quadrilateral is not convex, then either the three exterior points are saddles and the interior vertex an antisaddle or the exterior vertices are antisaddles and the interior vertex a saddle. Preliminaries. A real quadratic differential system is one of the form x = P(x,y), y = Q(x,y), where P(x, y) and Q(x, y) are real second degree polynomials. We assume that the curves P = 0 and Q = 0 meet at four distinct points of the affine plane, and then, the four points are simple critical points also known as elementary singular points of the system. That is because the condition of four distinct points ensures that the tangents to P = 0 and Q = 0 at each of them do not coincide, so that the determinant Px(xi,yi) x Qy(xi,yi) Py(xi,yi) x Qx(xi,yi) is different from zero, i = 1, 2, 3, 4. But this is the defining condition of an elementary singular point of the system and it is known that such point is either a saddle or an antisaddle: focus, center, node (see [2, Chapter 6, ?3]). The index of a saddle (angular variation of the field along a simple closed curve surrounding the singular point and no other) is -1, and antisaddles have index + 1. In the course of the proof we will need the fact that the sum of indices of a set of singular points is equal to the angular variation of the field along any simple closed curve surrounding those points and no others, also called the index of the curve with respect to the field (see [2, Appendix, ? 1]). These four critical points of the system are also base points of a pencil of conics consisting of P = 0 and the curves XP Q = 0, with X a real Received by the editors July 14, 1978 and in revised form, February 28, 1979. AMS (MOS) subject classifications (1970). Primary 34CO5 ? 1980 American Mathematical Society 0002-9939/80/0000-01 15/$01.75