6533b7ddfe1ef96bd1275475
RESEARCH PRODUCT
Unfolding of saddle-nodes and their Dulac time
Mariana SaavedraPavao MardešićJordi VilladelpratDavid Marínsubject
[ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]Block (permutation group theory)Dynamical Systems (math.DS)Space (mathematics)01 natural sciencesCombinatoricsQuadratic equationFOS: MathematicsMathematics - Dynamical Systems0101 mathematicsBifurcationSaddleMathematicsPeriod functionApplied MathematicsUnfolding of a saddle-node010102 general mathematics16. Peace & justice010101 applied mathematicsMSC: 34C07Asymptotic expansions34C07Node (circuits)Asymptotic expansionAnalysisdescription
Altres ajuts: UNAB10-4E-378, co-funded by ERDF "A way to build Europe" and by the French ANR-11-BS01-0009 STAAVF. 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 Theorem A and Theorem 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).
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2016-12-05 |