Unfoldings of saddle-nodes and their Dulac time

In this paper we study unfoldings of saddle-nodes and their Dulac time. By unfolding a saddle-node, saddles and nodes appear. In the first result (Theorem A) we give a uniform asymptotic expansion of the trajectories arriving at the node. Uniformity is with respect to all parameters including the unfolding parameter bringing the node to a saddle-node and a parameter belonging to a space of functions. In the second part, we apply this first result for proving a regularity result (Theorem B) on the Dulac time (time of Dulac map) of an unfolding of a saddle-node. This result is a building block in the study of bifurcations of critical periods in a neighborhood of a polycycle. Finally, we apply Theorems A and B to the study of critical periods of the Loud family of quadratic centers and we prove that no bifurcation occurs for certain values of the parameters (Theorem C). (C) 2016 Elsevier Inc. All rights reserved.