Nilpotence of orbits under monodromy and the length of Melnikov functions

Pavao Mardešić, Dmitry Novikov, Laura Ortiz-Bobadilla, Jessie Pontigo-Herrera

Research output: Contribution to journalArticlepeer-review

Abstract

Let F∈ℂ[x,y] be a polynomial, γ(z)∈π1(F−1(z)) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation dF+εω=0 of the integrable foliation given by F. We study different invariants: the orbit depth k, the nilpotence class n, the derivative length d associated with the couple (F,γ). These invariants bind the length ℓ of the first nonzero Melnikov function of the deformation dF+εω along γ. We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this behavior with the length of the corresponding Godbillon–Vey sequence. We formulate a conjecture motivated by the study of this example.

Original languageEnglish
Article number133017
JournalPhysica D: Nonlinear Phenomena
Volume427
DOIs
StatePublished - Dec 2021

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Fingerprint

Dive into the research topics of 'Nilpotence of orbits under monodromy and the length of Melnikov functions'. Together they form a unique fingerprint.

Cite this