Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres

Abstract

In this paper we consider the unfolding of saddle-node parametrized by with and in an open subset of and we study the Dulac time of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative tends to as uniformly on compact subsets of This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.