Unfoldings of saddle-nodes and their Dulac time [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Jordi Villadelprat
Universitat Rovira i Virgili


By unfolding a saddle-node, saddles and nodes appear. I will explain three different results. The first result (Theorem A) gives 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. This is used 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, I will explain how we apply Theorems A and B to the study of critical periods of the Loud family of quadratic centers and to prove that no bifurcation occurs for certain values of the parameters (Theorem C).

This is a joint work with P. Mardesic (Université de Bourgogne, France), D. Marín (UAB) and M. Saavedra (Universidad de Concepción, Chile)