The first nonzero Melnikov function for a family of good divides

  • Jessie Pontigo-HerreraEmail author
Original Paper


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.


Displacement function Melnikov function Limit cycles 

Mathematics Subject Classification

34M35 34C07 14D05 



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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