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arXiv · 1503.04629

Unfoldings of saddle-nodes and their Dulac time

Pavao Mardesić·David Marín·Mariana Saavedra·Jordi Villadelprat
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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).

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Pavao Mardesić, David Marín, Mariana Saavedra, Jordi Villadelprat. 2015-03-16. Unfoldings of saddle-nodes and their Dulac time. https://arxiv.org/abs/1503.04629

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