BIFURCATIONS OF CRITICAL PERIODS: CUBIC VECTOR FIELDS IN KAPTEYN'S NORMAL FORM

Abstract

In this paper, we study the local bifurcations of critical periods in the neighbourhood of a non-degenerate centre of a cubic system in Kapteyn's normal form. We find that this system has no isochronous centres. We show that at most three local critical periods bifurcate from a centre at the origin. Moreover there exist perturbations which lead to the maximum number of critical periods.