Abstract

The problems related to the Poincaré map often exhibit a similar formulation in terms of the time (or period) function associated to a continuum of periodic orbits. In this talk, parallel to the Melnikov method used to study the periodic orbits that persist after a perturbation of a centre, we present an intrinsic general formula for the derivative of the period function. This formula is obtained by exploiting the symmetries of a planar vector field $X$ having an isochronous centre, and it is applied to estimate the number of critical periods of a close vector field $X$_ε=X+ε Y having a centre. We will show examples of perturbations of quadratic and potential isochronous centres in which we use the first order of the perturbed period function