On the configurations of the singular points and their topological indices for the spatial quadratic polynomial differential systems

Abstract

Using the Euler-Jacobi formula there is a relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the polynomial differential systems x˙=P(x,y,z), y˙=Q(x,y,z), z˙=R(x,y,z) with degrees of P, Q and R equal to two when these systems have the maximum number of isolated singular points, i.e., 8 singular points. In other words we extend the well-known Berlinskii's Theorem for quadratic polynomial differential systems in the plane to the space.