Bifurcation of limit cycles from a centre in 4 in resonance 1:N

For every positive integer N 2 we consider the linear differential centre [image omitted] in 4 with eigenvalues i and Ni. We perturb this linear centre inside the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree N, i.e. [image omitted] where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree N. Then if 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.