Bifurcation of limit cycles from a 4-dimensional center in Rm in resonance 1:N

For every positive integer N⩾2 we consider the linear differential center x˙=Ax in Rm with eigenvalues ±i, ±Ni and 0 with multiplicity m−4. We perturb this linear center inside the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree N, i.e. x˙=Ax+εF(x) where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree N. 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 center. In particular, we give explicit upper bounds for the number of limit cycles.