Center problem and ν-cyclicity of polynomial zero-Hopf singularities with non-singular rotation axis

Abstract We consider three-dimensional polynomial families of vector fields parameterized by the admissible coefficients having a fixed zero-Hopf equilibrium and a non-singular rotation axis through it. We are interested in the periodic ν-orbits, that is, those orbits that makes a fixed arbitrary number ν (or a divisor of ν) of revolutions about the rotation axis and then returns to the initial point closing the orbit. We develop a Bautin-type method to study the ν-cyclicity of the equilibrium, that is, the maximum number of small-amplitude ν-limit cycles (isolated periodic ν-orbits) that can be made to bifurcate from the equilibrium by moving the parameters of the family restricted to some open semi-algebraic sets. The method uses branching theory based on Newton-Puiseux Theorem to get a finite number of an analytic reduced one-dimensional Poincare maps with associated Bautin ideal on certain Noetherian ring of rational functions on an extended parameter space. We derive global upper bounds on the number of bifurcated ν-limit cycles even in the infinite codimension case for which the perturbation of a local two-dimensional invariant manifold through the singularity completely foliated by periodic ν-orbits needs to be performed. A cubic normal form serves as an example of this procedure.