Universal Curves in the Center Problem for Abel Differential Equations

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.