Study of the period function of a two-parameter family of centers

In this paper we study the period function of ẍ = (1 x) p − (1 x) q , with p, q ∈ R and p > q. We prove three independent results. The first one establishes some regions in the parameter space where the corresponding center has a monotonous period function. This result extends the previous ones by Miyamoto and Yagasaki for the case q = 1. The second one deals with the bifurcation of critical periodic orbits from the center. The third one is addressed to the critical periodic orbits that bifurcate from the period annulus of each one of the three isochronous centers in the family when perturbed by means of a one-parameter deformation. These three results, together with the ones that we obtained previously on the issue, leads us to propose a conjectural bifurcation diagram for the global behaviour of the period function of the family.