Parametric Center-Focus Problem for Abel Equation

Abstract

The Abel differential equation $y^{\prime }=p(x)y^3+q(x)y^2$ 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 Poincaré Center-Focus problem for plane vector fields. Following [3,4,8,9] we say that Abel equation has a "parametric center" if for each $\epsilon \in \mathbb{C}$ the equation $y^{\prime }=p(x)y^3+\epsilon q(x)y^2$ 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 $°p,q\le 10$ 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.

This version is identical to the first one. The replacement is due to the fact that by mistake as a second version another paper was downnloaded. The paper was published by "Qual. Theory Dyn. Syst" in 2014

References in corpus (4)

• On rational functions orthogonal to all powers of a given rational function on a curve
• Algebraic Geometry of the Center-Focus problem for Abel Differential Equation
• On decompositions of trigonometric polynomials
• Weak and strong composition conditions for the Abel differential equation