6533b7d5fe1ef96bd1263d29
RESEARCH PRODUCT
Abelian integrals and limit cycles
Freddy DumortierRobert Roussariesubject
Abelian integralPure mathematicsApplied MathematicsMathematical analysisAbelian integralTwo-saddle cyclePlanar vector fieldsAsymptotic scale deformationCodimensionLimit cycleUpper and lower boundsPlanar vector fieldsymbols.namesakeLimit cyclesymbolsHamiltonian perturbationAbelian groupHamiltonian (quantum mechanics)BifurcationAnalysisMathematicsdescription
Abstract The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2006-08-01 | Journal of Differential Equations |