Abstract
© 2016 Elsevier Inc. 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).
Original language | English |
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Pages (from-to) | 6411-6436 |
Journal | Journal of Differential Equations |
Volume | 261 |
DOIs | |
Publication status | Published - 5 Dec 2016 |
Keywords
- Asymptotic expansions
- Period function
- Unfolding of a saddle-node