On the Chebyshev Property of Certain Abelian Integrals Near a Polycycle

Marín, D ^[1] ; Villadelprat, J
1. [1] Universitat Autònoma de Barcelona
  
  Universitat Autònoma de Barcelona
  
  Barcelona, España
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 17, Nº 1, 2018, págs. 261-270
Idioma: inglés
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Resumen
- Dumortier and Roussarie formulated in (Discrete Contin Dyn Syst 2:723–781, 2009) a conjecture concerning the Chebyshev property of a collection I0,I1,…,In of Abelian integrals arising from singular perturbation problems occurring in planar slow-fast systems. The aim of this note is to show the validity of this conjecture near the polycycle at the boundary of the family of ovals defining the Abelian integrals. As a corollary of this local result we get that the linear span ⟨I0,I1,…,In⟩ is Chebyshev with accuracy k=k(n).