Unfoldings of saddle-nodes and their Dulac time

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• 作者：P. Marde&scaron ; ić^a ; D. Marí ; n^b ; M. Saavedra^c ; J. Villadelprat^d ; ^{jordi.villadelprat@urv.cat" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34C07
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 December 2016
• 年：2016
• 卷：261
• 期：11
• 页码：6411-6436
• 全文大小：445 K

文摘

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  and  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).