  This article is a survey on recent contributions to an effective version of
Bautin's theory about the bifurcation of periodic orbits (limit cycles). The
analysis of Hopf bifurcations of higher order is possible by use of the return
mapping. Explicit estimates of the size of the domain on which the number of
limit cycles is controlled are important in many applications of bifurcation
theory (for instance in Biology and Physiology).
