Abstract

Consider a family of planar systems $x\prime =X(x,\epsilon )$ having a center at the origin and assume that for $\epsilon =0$ they have an isocronous center. Firstly, we give an explicit formula for the first order term in $\epsilon$ of the derivative of the period function. We apply this formula to prove that, in first order in $\epsilon$, at most one critical period bifurcates from the isochronous quadratic systems when we perturb them inside the class of quadratic reversible centers. Moreover necessary and sufficient condition for the existence of such critical period are explicitly given. This is a joint work with A. Gasull.