Perturbed rigidly isochronous centers and their critical periods

This paper investigates the bifurcation of critical periods from a cubic rigidly isochronous center under any small polynomial perturbations of degree n. It proves that for n=3,4 and 5, there are at most 2 and 4 critical periods induced by periodic orbits of the unperturbed cubic system respectively, and in each case this upper bound is sharp. Moreover, for any n>5, there are at most [[formula presented]] critical periods induced by periodic orbits of the unperturbed cubic system. An example is given to show that the upper bound in the case of n=11 can be reached.