Darboux integrable system with a triple point and pseudo-abelian integrals

2016/11/03 by Aymen Braghtha, Braghtha, Aymen
Mathematics · Physics and Astronomy · #34C07 #34C08 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Waves and Solitons

paper · doi:10.48550/arxiv.1611.04533

openalex publication_date 2016/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the degeneracies of the third type. More exact, the perturbations of the Darboux integrable foliation with a triple point, i.e. the case where three of the curves \P_i = 0\ meet at one point, are considered. Assuming that this is the only non-genericity, we prove that the number of zeros of the corresponding pseudo-abelian integrals is bounded uniformly for close Darboux integrable foliations. Let F denote the foliation with triple point (assume it to be at the origin), and let F_λ= \M_λdH_λ\over H_λ = 0\, M_λ is a integrating factor, be the close foliation. The main problem is that F_λ can have a small nest of cycles which shrinks to the origin as λ→ 0. A particular case of this situation, namely H_λ= (x -λ)^ε(y - x)^ε₊ (y + x)^ε_-Δ with Δ non-vanishing at the origin (and generic in appropriate sense).

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