TY - JOUR

T1 - Parametric Center-Focus Problem for Abel Equation

AU - Briskin, M.

AU - Pakovich, F.

AU - Yomdin, Y.

N1 - Funding Information:
This research was supported by the ISF, Grants No. 639/09 and 779/13, and by the Minerva Foundation.
Publisher Copyright:
© 2014, Springer Basel.

PY - 2014/10/31

Y1 - 2014/10/31

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84930797464&partnerID=8YFLogxK

U2 - 10.1007/s12346-014-0118-8

DO - 10.1007/s12346-014-0118-8

M3 - Article

AN - SCOPUS:84930797464

SN - 1575-5460

VL - 13

SP - 289

EP - 303

JO - Qualitative Theory of Dynamical Systems

JF - Qualitative Theory of Dynamical Systems

IS - 2

ER -