Center conditions. II: Parametric and model center problems
The paper is concerned with the following initial value problem for Abel's equation \[ y'=p(x)y^2+ q(x)y^3,\quad y(0)=y_0, \tag{1} \] where \(p\) and \(q\) are polynomials of the degree \(d_1\) and \(d_2\), respectively. A center condition for the IVP (1) is known as \(y_0=y(0) \equiv y(1)\) for any solution \(y(x)\) to (1). Introducing a parameter in the IVP (1) \[ y'=p(x) y^2+ \varepsilon q(x)y^3, \quad y(0)=y_0, \tag{2} \] where \(p\) and \(q\) are as above and \(\varepsilon\) is a complex number, the authors formulate a parametric center condition for the IVP (2) as \(y(\varepsilon,0) \equiv y(\varepsilon,1)\) for any solution \(y(\varepsilon,x)\) to (2). It is shown that the parametric center condition implies that all the moments \(m_k(1)\) vanish, where \[ m_k(x)= \int^x_0 P^k (t)q(t)dt, \quad P(z)=\int^x_0 p(t)dt. \] The paper is a continuation of the papers by the same authors [Nonlinearity 11, No. 3, 431-443 (1998; Zbl 0905.34051)] and Center conditions, compositions of polynomials and moments on algebraic curves, Ergodic Theory Dyn. Syst. 19, No. 5, 1201-1220 (1999; Zbl 0990.34017)], and one of its main purposes is to prove a conjecture stated in the latter paper for some special cases developing moment techniques and an algebraic analysis of the \(m_k(x)\) and their representation in a canonical form. Finally, a combinatorial approach to the problem of stabilization of the moment sequence is briefly discussed.
scientific article; zbMATH DE number 1506582
scientific article