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