Universal Curves in the Center Problem for Abel Differential Equations

TL;DR

This paper investigates the properties of universal curves in the center problem for Abel differential equations, providing algebraic characterizations, explicit examples, and approximation methods for such curves.

Contribution

It introduces a new algebraic framework for universal curves, offers explicit examples, and develops approximation techniques for Lipschitz triangulable curves.

Findings

Algebraic description of universal curves via fundamental group homomorphism

Explicit examples of universal curves provided

Approximation of Lipschitz curves by universal curves achieved

Abstract

We study the center problem for the class $\mathcal E_\Gamma$ of Abel differential equations $\frac{dv}{dt}=a_1 v^2+a_2 v^3$, $a_1,a_2\in L^\infty ([0,T])$, such that images of Lipschitz paths $\tilde A:=\bigl(\int_0^\cdot a_1(s)ds, \int_0^\cdot a_2(s)ds\bigr): [0,T]\rightarrow\mathbb R^2$ belong to a fixed compact rectifiable curve $\Gamma$. Such a curve is called universal if whenever an equation in $\mathcal E_\Gamma$ has center on $[0,T]$, this center must be universal, i.e. all iterated integrals in coefficients $a_1, a_2$ of this equation must vanish. We investigate some basic properties of universal curves. Our main results include an algebraic description of a universal curve in terms of a certain homomorphism of its fundamental group into the group of locally convergent invertible power series with product being the composition of series, explicit examples of universal curves and approximation of Lipschitz triangulable curves by universal ones.