Search results for "Abelian integrals"

showing 4 items of 4 documents

Darboux integrable system with a triple point and pseudo-abelian integrals

2016

We study pseudo-abelian integrals associated with polynomial perturbations of Dar-boux integrable system with a triple point. Under some assumptions we prove the local boundedness of the number of their zeros. Assuming that this is the only non-genericity, we prove that the number of zeros of the corresponding pseudo-abelian integrals is bounded uniformly for nearby Darboux integrable foliations.

0209 industrial biotechnologyPure mathematicsControl and OptimizationIntegrable systemTriple pointAbelian integrals[ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS]Darboux integrability[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]Dynamical Systems (math.DS)02 engineering and technologyType (model theory)01 natural sciencesIntegrating factor020901 industrial engineering & automationFOS: MathematicsLimit Cycle0101 mathematicsAbelian groupMathematics - Dynamical Systems34C07 34C08MathematicsNumerical AnalysisAlgebra and Number Theory010102 general mathematicsMathematical analysisLimit cyclesMathematics Subject ClassificationControl and Systems EngineeringBounded functionFoliation (geology)
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Nilpotence of orbits under monodromy and the length of Melnikov functions

2021

Abstract Let F ∈ ℂ [ x , y ] be a polynomial, γ ( z ) ∈ π 1 ( F − 1 ( z ) ) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation d F + e ω = 0 of the integrable foliation given by F . We study different invariants: the orbit depth k , the nilpotence class n , the derivative length d associated with the couple ( F , γ ) . These invariants bind the length l of the first nonzero Melnikov function of the deformation d F + e ω along γ . We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this b…

PhysicsPure mathematicsSequencePolynomialConjectureMelnikov functionAbelian integrals010102 general mathematicsStatistical and Nonlinear PhysicsIterated integralsCondensed Matter Physics01 natural sciencesNilpotence classFoliationDisplacement functionLimit cyclesMonodromySimple (abstract algebra)[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]Product (mathematics)0103 physical sciences010307 mathematical physics0101 mathematicsOrbit (control theory)ComputingMilieux_MISCELLANEOUS
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Darboux systems with a cusp point and pseudo-abelian integrals

2018

International audience; We study pseudo-abelian integrals associated with polynomial deformations of Darboux systems having a cuspidal singularity. Under some genericity hypothesis we provide locally uniform boundedness of on the number of their zeros.

[ MATH ] Mathematics [math]Cusp (singularity)Pure mathematicsPolynomialApplied Mathematics[ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS]010102 general mathematics[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]Darboux integrability[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]Pseudo-abelian integrals[MATH] Mathematics [math]01 natural sciences010101 applied mathematicsLimit cyclesSingularityUniform boundednessPoint (geometry)First integral0101 mathematicsAbelian groupMSC : 34C07 ; 34C08[MATH]Mathematics [math]AnalysisMathematics
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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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