On the critical periods of perturbed isochronous centers

Consider a family of planar systems having a center at the origin and assume that for ε=0 they have an isochronous center. Firstly, we give an explicit formula for the first order term in ε of the derivative of the period function. We apply this formula to prove that, up to first order in ε, at most one critical period bifurcates from the periodic orbits of isochronous quadratic systems when we perturb them inside the class of quadratic reversible centers. Moreover necessary and sufficient conditions for the existence of this critical period are explicitly given. From the tools developed in this paper we also provide a new characterization of planar isochronous centers.