Planar real polynomial differential systems of degree $n> 3$ having a weak focus of high order

Abstract

We construct planar polynomial differential systems of even (respectively odd) degree n > 3, of the form linear plus a nonlinear homogeneous part of degree n having a weak focus of order n2 − 1 (respectively n2−1/2) at the origin. As far as we know this provides the highest order known until now for a weak focus of a polynomial differential system of arbitrary degree n.