Parametric Center-Focus Problem for Abel Equation

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 Poincaré Center-Focus problem for plane vector fields. Following Briskin et al. (Ergodic Theory Dyn Syst 19(5):1201–1220, 1999; Isr J Math 118:61–82, 2000); Cima et al. (Qual Theory Dyn Syst 11(1):19–37, 2012; J Math Anal Appl 398(2):477–486, 2013) we say that Abel equation has a “parametric center” if for each $$\epsilon \in \mathbb C$$ the equation $$y'=p(x)y^3 + \epsilon q(x) y^2$$ has a center. In the present paper we use recent results of Briskin et al. (Algebraic Geometry of the Center-Focus problem for Abel differential equations, arXiv:1211.1296 , 2012); Pakovich (Comp Math 149:705–728, 2013) 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 $$\deg p,q \le 10$$ parametric center is equivalent to the so-called “Composition Condition” (CC) (Alwash and Lloyd in Proc R Soc Edinburgh 105A:129–152, 1987; Briskin et al. Ergodic Theory Dyn Syst 19(5):1201–1220, 1999) on $$p,q$$ . Second, we study trigonometric Abel equation, and provide a series of examples, generalizing a recent remarkable example given in Cima et al. (Qual Theory Dyn Syst 11(1):19–37, 2012), where certain moments of $$p,q$$ vanish while (CC) is violated.