Monotonous Period Function for Equivariant Differential Equations with Homogeneous Nonlinearities

Abstract

We prove that the period function of the center at the origin of the Zk-equivariant differential equation z˙=iz+a(zz¯)nzk+1,a≠0, is monotonous decreasing for all n and k positive integers, solving a conjecture about them. We show this result as corollary of proving that the period function of the center at the origin of a sub-family of the reversible quadratic centers is monotonous decreasing as well.