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 [3,4,8,9] we say that Abel equation has a "parametric center" if for each ǫ ∈ C the equation y ′ = p(x)y 3 + ǫq(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 deg p, q ≤ 10 parametric center is equivalent to the so-called "Composition Condition" (CC) ([2, 3]) 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.----------------

Solution of the parametric center problem for the Abel differential equation

“…Theorem 2.1 ([17]) Let P, Q be non-constant complex polynomials and a, b distinct complex numbers such that equalities (7) hold. Then, either Q is a reducible solution of (7), or there exist complex polynomials…”

“…Posed for the first time in the series of papers [3], [4], [5], this problem turned out to be very constructive and resulted in a whole area of new ideas and methods related to the so called "polynomial moment problem" (see the discussion below). However, in its full generality the parametric center problem remained unsolved (see the recent paper [7] for the state of the art), and the goal of this paper is to fill this gap. Our main result is the following theorem.…”

“…[3], [4] [5], [10], [11], [12], [13], [14], [15], [16], [17], [19]). Again, the composition condition (4) is sufficient for equalities (7) to be satisfied although in general is not necessary ( [11]). A complete solution of the polynomial moment problem was obtained in the recent papers [15], [17].…”

“…A complete solution of the polynomial moment problem was obtained in the recent papers [15], [17]. Namely, it was shown in [15] that if polynomials P, Q satisfy (7), then there exist polynomials Q j such that Q = j Q j and P (x) = P j (W j (x)), Q j (x) = V j (W j (x)), W j (a) = W j (b) (8) for some polynomials P j (z), V j (z), W j (z). Moreover, in [17] polynomial solutions of (7) were described in an explicit form (see Section 2 below).…”

“…Namely, it was shown in [15] that if polynomials P, Q satisfy (7), then there exist polynomials Q j such that Q = j Q j and P (x) = P j (W j (x)), Q j (x) = V j (W j (x)), W j (a) = W j (b) (8) for some polynomials P j (z), V j (z), W j (z). Moreover, in [17] polynomial solutions of (7) were described in an explicit form (see Section 2 below). In this paper we apply results of [17] to each of the two systems appeared in (6) separately and show that the restrictions obtained imply that any solution P, Q of the mixed polynomial moment problem satisfy composition condition (4).…”