Parametric Center-Focus Problem for Abel Equation

The Abel differential equation y^′=p(x)^y3+q(x)^y2 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 (Formula presented.) the equation (Formula presented.) 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 (Formula presented.) 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.