On the center problem for ordinary differential equations

Abstract

We study the moments finiteness problem for the class of Lipschitz maps F : [a, b] → R n with images in a compact Lipschitz triangulable curve Γ. We apply the obtained results to the center problem for ODEs describing in some cases (including equations with analytic coefficients) the set of universal centers of such equations by vanishing of finitely many moments from their coefficients.

Cauchy-type integrals of algebraic functions

“…This conjecture has been verified for small degrees of p and q and in many special cases in [9][10][11][12][13][14][15][16][17][18][19]21,58,61].…”

“…Although the Center and the Hilbert problems for equation (C) on an interval (or in general on a non-closed curve) do not correspond directly to the classical setting, they presents an interest by their own and they have been intensively investigated in [3][4][5][19][20][21][22]25,36,37,54,58] and in many other publications. It is a general belief that the Center and Hilbert problems for a polynomial Abel equation (C) on the interval present all the main difficulties of the classical ones while possibly simplifying essential technical details.…”

“…Recently in [9][10][11][12][13][14][15][16][17][18]21,47,61] the Center problem for both the trigonometric and the polynomial Abel equations has been related to the problem of the vanishing of certain generalized moments on one hand and to the Composition algebra of univariate analytic functions on the other. This approach has been further developed in [19,31,58].…”

“…The condition (PCC) has been introduced and intensively studied in [9][10][11][12][13][14][15][16][17][18]21,47,61] (see also [19,22,31,58]). There is a growing evidence supporting the major role played by the Polynomial Composition condition (and in general, by the polynomial Composition Algebra) in the structure of the Center conditions for the polynomial Abel equation (C).…”

Some Algebraic Aspects of the Center Problem for Ordinary Differential Equations

“…In this paper we present some algebraic aspects of the center problem for equations (1.2) based on the results of papers [8][9][10][11][12]. We also formulate certain open problems in this area.…”