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 language | English |
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Article number | 133017 |
Journal | Physica D: Nonlinear Phenomena |
Volume | 427 |
DOIs | |
State | Published - Dec 2021 |
All Science Journal Classification (ASJC) codes
- Condensed Matter Physics
- Applied Mathematics
- Statistical and Nonlinear Physics
- Mathematical Physics