Unfoldings of saddle-nodes and their Dulac time

TL;DR

This paper investigates the unfolding of saddle-nodes, establishing uniform regularity of orbits and derivatives, and applies these results to analyze Dulac times and bifurcations of critical periods in quadratic centers.

Contribution

It provides new regularity results for unfoldings of saddle-nodes and their Dulac times, aiding the understanding of bifurcations in dynamical systems.

Findings

Uniform regularity of orbits and derivatives near saddle-nodes

Regularity of Dulac time in unfoldings of saddle-nodes

No bifurcation occurs for certain parameters in the Loud family of quadratic centers

Abstract

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 prove uniform regularity by which orbits and their derivatives arrive at a 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 neighbourhood 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).