Cyclicity of polynomial nondegenerate centers on center manifolds

We consider polynomial families of ordinary differential equations on R3, parametrized by the admissible coefficients, for which the origin is an isolated singularity at which the linear part of the system has one non-zero real and two purely imaginary eigenvalues. We derive theorems that bound the maximum number of limit cycles within the center manifold that can bifurcate, under arbitrarily small perturbation of the coefficients, from any center at the origin. The bounds are global in that they apply to the system corresponding to any point on any irreducible component of the center variety in the space of parameters. We also derive theorems for such bounds when attention is confined to a single irreducible component of the center variety.