Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation

2015/04/09 by Dmitry Batenkov, Gal Binyamini, Batenkov, Dmitry +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Quantum chaos and dynamical systems #math.CA

paper · pdf · doi:10.48550/arxiv.1504.02208

arxiv created 2015/04/09 · openalex publication_date 2015/04/09 · arxiv updated 2015/04/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/04

Abstract

Given two polynomials P,q we consider the following question: "how large can the index of the first non-zero moment m_k=∫_a^b P^k q be, assuming the sequence is not identically zero?". The answer K to this question is known as the moment Bautin index, and we provide the first general upper bound: K\leqslant 2+deg q+3(deg P-1)². The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions. The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation y'=py²+ε qy³ for p,q polynomials and ε ≪ 1. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed 5+deg q+3deg² p. This is the first such bound depending solely on the degrees of the Abel equation.

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