On the configurations of the singular points and their topological indices for the spatial quadratic polynomial differential systems
Consider a real planar quadratic polynomial differential system \[ \dot x = P(x,y), \ \ \dot y = Q(x,y) \] that has exactly 4 real singular points. The well-known Berlinskii Theorem states that if the quadrilater formed by these points is convex, then two opposite singular points are anti-saddles (i.e., nodes, foci or centers) and the other two are saddles. If this quadrilater is not convex, then either the three exterior vertices are saddles and the interior vertex is an anti-saddle or the exterior vertices are anti-saddles and the interior vertex is a saddle. The authors extend the Berlinskii Theorem for quadratic polynomial differential systems in the plane to the space. They deal with differential systems \[ \dot x = P(x,y,z), \ \ \dot y = Q(x,y,z), \ \ \dot z = R(x,y,z), \] where \(P,Q,R\) are real polynomials of degrees 2. Assuming that the above system has 8 real singular points, the authors obtain all the configurations of the singular points and their topological indices.
scientific article; zbMATH DE number 7260117