0000000000419693

AUTHOR

Mariana Saavedra

showing 3 related works from this author

Unfolding of saddle-nodes and their Dulac time

2016

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 b…

[ 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 expansionAnalysis
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Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles

2007

We call Poincare time the time associated to the Poincar6 (or first return) map of a vector field. In this paper we prove the non-accumulation of isolated critical points of the Poincare time T on hyperbolic polycycles of polynomial vector fields. The result is obtained by proving that the Poincare time of a hyperbolic polycycle either has an unbounded principal part or is an almost regular function. The result relies heavily on the proof of Il'yashenko's theorem on non-accumulation of limit cycles on hyperbolic polycycles.

Critical period; finiteness; non-accumulation; quasi-analyticity; Dulac problem.Applied MathematicsGeneral MathematicsLimit cycleMathematical analysisHyperbolic manifoldPrincipal partUltraparallel theoremVector fieldRelatively hyperbolic groupCritical point (mathematics)Hyperbolic equilibrium pointMathematicsProceedings of the American Mathematical Society
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Principal part of multi-parameter displacement functions

2012

This paper deals with a perturbation problem from a period annulus, for an analytic Hamiltonian system [J.-P. Françoise, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 87–96 ; L. Gavrilov, Ann. Fac. Sci. Toulouse Math. (6) 14(2005), no. 4, 663–682. The authors consider the planar polynomial multi-parameter deformations and determine the coefficients in the expansion of the displacement function generated on a transversal section to the period annulus. Their first result gives a generalization to the Françoise algorithm for a one-parameter family, following [J.-P. Françoise and M. Pelletier, J. Dyn. Control Syst. 12 (2006), no. 3, 357–369. The second result expresses the principal terms in …

MonomialMathematics(all)Abelian integralsGeneral MathematicsHamiltonian system; perturbation; triangle centerMathematical analysisIterated integralsStandard basisMelnikov functionsDisplacement functionLimit cyclessymbols.namesakePlanarIterated integralsBautin idealBounded functionsymbolsPrincipal partVector fieldHamiltonian (quantum mechanics)Multi parameterMathematicsBulletin des Sciences Mathématiques
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