Bifurcation of critical periods from the reversible rigidly isochronous centers

In this paper we study bifurcation of critical periods by perturbing a nonlinear vector field containing mm-th degree homogeneous terms with a rigidly isochronous center at the origin. First, we give expressions of period bifurcation functions in the form of integrals, and then study the number of critical periods for cases of m=1,2,3m=1,2,3 respectively.