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Journal of Dynamical and Control Systems

, Volume 23, Issue 3, pp 597–622 | Cite as

Tangential Center Problem for a Family of Non-generic Hamiltonians

  • Jessie Pontigo-HerreraEmail author
Article

Abstract

The tangential center problem was solved by Yu. S. Ilyashenko in the generic case Mat Sbornik (New Series), 78, 120, 3,360–373, (1969). With the aim of having well-understood models of non-generic Hamiltonians, we consider here a family of non-generic Hamiltonians, whose Hamiltonian is of the form \(F=\prod f_{j}\), where f j are real polynomials of degree ≥ 1. For this family, the genericity assumption of transversality at infinity fails and the coincidence of the critical values for different critical points is allowed. We consider some geometric conditions on these polynomials in order to compute the orbit under monodromy of their vanishing cycles. Under those conditions, we provide a solution of the tangential center problem for this family.

Keywords

Abelian integrals Tangential center problem Monodromy 

Mathematics Subject Classification (2010)

34M35 34C08 14D05 

Notes

Acknowledgments

I am grateful to Laura Ortiz, Pavao Mardesic and Dmitry Novikov for numerous helpful discussions and suggestions. I am also grateful to the referee for useful and accurate remarks.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de México (UNAM)Distrito FederalMéxico
  2. 2.Institut de Mathématiques de Bourgogne, U.M.R. 5584 du C.N.R.S.Université de BourgogneDijon CedexFrance

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