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

TL;DR

This paper establishes a new upper bound for the moment Bautin index related to polynomial Abel equations, which helps limit the number of periodic solutions near zero based on polynomial degrees.

Contribution

The paper provides the first general upper bound for the moment Bautin index using qualitative analysis of linear ODEs and applies it to bound periodic solutions in polynomial Abel equations.

Findings

Bound K ≤ 2 + deg q + 3(deg P - 1)^2 for the moment Bautin index.

Number of periodic solutions near zero does not exceed 5 + deg q + 3 deg^2 p.

First bound depending solely on the degrees of the Abel equation.

Abstract

Given two polynomials $P,q$ we consider the following question: "how large can the index of the first non-zero moment $\tilde{m}_k=\int_a^b P^k q$ 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: $K\leqslant 2+\mathrm{deg} q+3(\mathrm{deg} P-1)^2$. 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 $y'=py^2+\varepsilon qy^3$ for $p,q$ polynomials and $\varepsilon \ll 1$. 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 $5+\mathrm{deg} q+3\mathrm{deg}^2 p$. This is the first such bound depending solely on the degrees of the Abel equation.