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.
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This research was supported by the ISF, Grants No. 639/09 and 779/13, and by the Minerva Foundation.
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Briskin, M., Pakovich, F. & Yomdin, Y. Parametric Center-Focus Problem for Abel Equation. Qual. Theory Dyn. Syst. 13, 289–303 (2014). https://doi.org/10.1007/s12346-014-0118-8
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DOI: https://doi.org/10.1007/s12346-014-0118-8