Limit cycles from a four-dimensional centre in ℝ m in resonance p : q

Abstract

Given positive coprime integers p and q, we consider the linear differential centre in ℝ m with eigenvalues ±pi, ±qi and 0 with multiplicity m − 4. We perturb this linear centre in the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree p + q − 1, i.e. , where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree p + q − 1. When the displacement function of order ϵ of the perturbed system is not identically zero, we study the maximal number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.