Bifurcation of critical periods for plane vector fields

A bifurcation problem in families of plane analytic vector fields which have a nondegenerate center at the origin for all values of a parameter λ ∈ R N { {R}N} is studied. In particular, for such a family, the period function (ξ, λ) ↦ P (ξ, λ) (, ) P (, ) is defined; it assigns the minimum period to each member of the continuous band of periodic orbits (parametrized by ξ ∈ R {R}) surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with λ as bifurcation parameter. Generally, if the function ρ, given by ξ ↦ P (ξ, λ ∗) − P (0, λ ∗) P (, _) - P (0, _), vanishes to order 2 k 2k at the origin, then it is shown that the period function, after a perturbation of λ ∗