Parametric Center-Focus Problem for Abel Equation

TL;DR

This paper investigates the parametric center problem for Abel differential equations, establishing that for polynomial cases up to degree 10, a parametric center implies a specific composition condition, and explores trigonometric variants with examples.

Contribution

It proves that parametric centers in polynomial Abel equations up to degree 10 are equivalent to the Composition Condition, and extends analysis to trigonometric Abel equations with new examples.

Findings

Parametric center implies Composition Condition for degrees ≤ 10.

For polynomial Abel equations, parametric center is characterized by strong composition restrictions.

Constructed examples of trigonometric Abel equations with specific properties.

Abstract

The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with meromorphic coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abel equation is analogous to the classical Poincar\'e Center-Focus problem for plane vector fields. Following [3,4,8,9] we say that Abel equation has a "parametric center" if for each $\varepsilon \in \mathbb C$ the equation $y'=p(x)y^3 + \varepsilon q(x) y^2$ has a center. In the present paper we use recent results of [15,6} to show show that for a polynomial Abel equation parametric center implies strong "composition" restriction on $p$ and $q$. In particular, we show that for $\deg p,q \leq 10$ parametric center is equivalent to the so-called "Composition Condition" (CC) on $p,q$. Second, we study trigonometric Abel equation, and provide a series of examples, generalizing a recent remarkable example given in [8], where certain moments of $p,q$ vanish while (CC) is violated.