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The first nonzero Melnikov function for a family of good divides

  • Jessie Pontigo-HerreraEmail author
Original Paper
  • 4 Downloads

Abstract

In this paper we study polynomial Hamiltonian systems \(dF=0\) in the plane and their deformations \(dF+\epsilon \omega =0\), where \(\omega \) is a polynomial 1-form. We consider the first nonzero Melnikov function, \(M_{\mu }\), of the displacement function \(\Delta (t,\epsilon )=\sum _{j=\mu }^{\infty }\epsilon ^{j}M_{j}(t)\), along a cycle \(\gamma (t)\) in \(F^{-1}(t)\). It is known, that in the generic case \(M_{\mu }\) is an abelian integral (Françoise in Ergod Theory Dyn Syst 16(1):87–96, 1996; Ilyashenko in Mat Sb (N.S.) 78(120):360–373, 1969), and an iterated integral of length at most \(\mu \) in general (Gavrilov in Ann Fac Sci Toulouse Math (6) 14(4):663–682, 2005). Here we study linear deformations of a family of non-generic Hamiltonians systems \(dF=0\), where \(F=\prod _{j=1}^rf_j\in {{\mathbb {R}}}[x,y]\), with \(f_j=f_{1j}^{n_j}+g_j\), \(n_j\in {{\mathbb {N}}}\), for \(f_{1j}\)\((j=1,\ldots ,r)\) pairwise linearly independent polynomials of degree one, and \(g_j\) a polynomial of degree smaller than \(n_j\) (Pontigo-Herrera in J Dyn Control Syst 23(3):597–622, 2017). We also assume some geometric properties on F; namely, \(\overline{F^{-1}(0)}\) defines a good divide with r branches in \({{\mathbb {R}}}{{\mathbb {P}}}^2\) (where only the zero critical level can have more than one critical point) and F has good multiplicity at infinity (A’Campo in Math Ann 213:1–32, 1975; Pontigo-Herrera in J Dyn Control Syst 23(3):597–622, 2017). We denote this family by \({\mathcal {F}}_r({{\mathbb {R}}})\). We prove that for polynomials in \({\mathcal {F}}_r({{\mathbb {R}}})\), the first nonzero Melnikov function of their deformations are iterated integrals of length at most two.

Keywords

Displacement function Melnikov function Limit cycles 

Mathematics Subject Classification

34M35 34C07 14D05 

Notes

Acknowledgements

I would like to express my deepest gratitude to Laura Ortiz-Bobadilla, Pavao Mardešić and Dmitry Novikov for sharing with me their time and insights in so many discussions, suggestions and corrections. I also thank the reviewers for their valuable comments. The paper was written while the author was a postdoctoral fellow in Weizmann Institute of Science. I thank Weizmann Institute for all the support and facilities provided during the elaboration of this work.

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceWeizmann Institute of ScienceRehovotIsrael
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de Mexico (UNAM)Mexico CityMexico

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