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

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., ẋ = Ax+εF(x), 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 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. © 2012 Copyright Taylor and Francis Group, LLC.