0000000000419694

AUTHOR

Jordi Villadelprat

showing 4 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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The period function of reversible quadratic centers

2006

Abstract In this paper we investigate the bifurcation diagram of the period function associated to a family of reversible quadratic centers, namely the dehomogenized Loud's systems. The local bifurcation diagram of the period function at the center is fully understood using the results of Chicone and Jacobs [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486]. Most of the present paper deals with the local bifurcation diagram at the polycycle that bounds the period annulus of the center. The techniques that we use here are different from the ones in [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vecto…

Period-doubling bifurcationTranscritical bifurcationcenterApplied MathematicsMathematical analysisSaddle-node bifurcationInfinite-period bifurcationParameter spaceBifurcation diagramAsymptotic expansionAnalysisBifurcationMathematicsJournal of Differential Equations
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The index of stable critical points

2002

Abstract In this paper we show that in dimension greater or equal than 3 the index of a stable critical point can be any integer. More concretely, given any k∈ Z and n⩾3 we construct a C ∞ vector field on R n with a unique critical point which is stable (in positive and negative time) and has index equal to k. This result extends previous ones on the index of stable critical points.

CombinatoricsVector fieldPlug constructionIsolated critical pointVector fieldGeometry and TopologyTopologyStabilityCritical point (mathematics)MathematicsIndexTopology and its Applications
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On the time function of the Dulac map for families of meromorphic vector fields

2003

Given an analytic family of vector fields in Bbb R2 having a saddle point, we study the asymptotic development of the time function along the union of the two separatrices. We obtain a result (depending uniformly on the parameters) which we apply to investigate the bifurcation of critical periods of quadratic centres.

Differential equationApplied MathematicsMathematical analysisGeneral Physics and AstronomyStatistical and Nonlinear PhysicsQuadratic equationSaddle pointtime-map; quadratic centresDevelopment (differential geometry)Vector fieldAsymptotic expansionMathematical PhysicsBifurcationMathematicsMeromorphic functionNonlinearity
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