Unfoldings of saddle-nodes and their Dulac time

作者：Mardesic P; Marin D; Saavedra M; Villadelprat J^*

来源：Journal of Differential Equations, 2016, 261(11): 6411-6436.

DOI：10.1016/j.jde.2016.08.040

• Period function
• Unfolding of a saddle-node
• Asymptotic expansions

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).

• 出版日期2016-12-5