A counterexample to the composition condition conjecture for polynomial Abel differential equations

Abstract

The Polynomial Abel differential equations are considered a model problem for the classical Poincaré center-focus problem for planar polynomial systems of ordinary differential equations. Last decades several works pointed out that all the centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.2010 Mathematics Subject Classification. Primary 34C25. Secondary 34C07.

Two lectures on the center-focus problem for the equation $\frac{dy}{dx} + \sum_{i=1}^n a_i(x) y^i = 0, 0\leq x \leq 1$ where $a_i$ are polynomials

“…In this section we show that there are scalar Abel equations with a non-universal center (not related to a "pull back"). These equations have a Darboux type first integral, and among them we find the recent counter-example to the Composition Conjecture mentioned above, found by Giné, Grau and Santallusia [16].…”

“…Incidentally, Liénard equations with a Darboux type first integral will produce counter-examples to the Composition Conjecture, which is the subject of the present section. We explain in this context the recent counter-example of Giné, Grau and Santallusia [16].…”

“…via a suitable polynomial map ξ = ξ(x) having the property ξ(x 0 ) = ξ(x 1 ). Not all centers of (13) are universal, as discovered recently in [16]. For a further use, note that an obvious consequence from F (x, y) ≡ y is that…”

“…However, after the substitution 1 → 1/y, the above Liénard equation is equivalent to the Abel equation ( 54) dy dx = p(x)y 2 + q(x)y 3 with Darboux type first integral 16]). The Abel equation (54) has a center at y = 0 along the interval [−1, 1] but this center is not universal.…”

Infinitesimal Center Problem on zero cycles and the composition conjecture

“…The composition condition defines all the irreducible components of the center variety in many families of Abel equations [14], in some planar systems (see [27,28]), and accounts for most of the irreducible components when studying the tangential centers of the Abel equation at infinity [12]. However, not all centers satisfy the compostion conjecture in the trigonometric Abel equation [4], or even the polynomial Abel equation [22].…”