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The problems related to the Poincaré map often exhibit a similar formulation in terms of the time (orperiod) function associated to a continuum of periodic orbits. In this paper, 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 Lie symmetries of a planar vector fieldX having an isochronous centre, and it is applied to estimate the number of critical periods of a “close” vector fieldX ∈=X+∈Y having a centre.

Period function for perturbed isochronous centres