Unfoldings of saddle-nodes and their Dulac time

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 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).
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Contributeur : Imb - Université de Bourgogne <>
Soumis le : mardi 6 décembre 2016 - 15:04:27
Dernière modification le : vendredi 8 juin 2018 - 14:50:07

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Pavao Mardešić, David Marìn, M. Saavedra, Jordi Villadelprat. Unfoldings of saddle-nodes and their Dulac time. Journal of Differential Equations, Elsevier, 2016, 261 (11), pp. 6411-6436 ⟨10.1016/j.jde.2016.08.040 ⟩. ⟨hal-01410190⟩



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