Asymptotic expansion of the period function

Let P be a not necessarily bounded polycycle of an analytic vector field on an open set of the plane. Suppose that the singularities which appear after desingularization of the vertices of P are formally linearizable. Consider the function T defined by the return time near P. It is shown that the function T and its derivative T′ have asymptotic expansions in and . It is also shown that under some other conditions imposed on the polycycle vertices, the asymptotic expansions of T and T′ converge absolutely and uniformly to these functions, respectively. These results are applied to the polycycles of the analytic vector fields which have a Darboux first integral. In particular, it is obtained that if P is a polycycle of a Hamiltonian vector field with an analytic (polynomial if P is unbounded) Hamiltonian function, T is a nonoscillating function. Another application concerns the nilpotent centers or focus, since the singularities which appear after desingularization of such a singularity have analytic first integrals.