2014/07/01 by Fedor Pakovich, Pakovich, Fedor
Mathematics · #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CA #math.DS
paper · pdf · doi:10.48550/arxiv.1407.0150
arxiv created 2014/07/01 · arxiv updated 2014/07/02
The Abel differential equation y'=p(x)y2+q(x)y3 with p,q∈ \mathbb R[x] is said to have a center on a segment [a,b] if all its solutions, with the initial value y(a) small enough, satisfy the condition y(b)=y(a). The problem of description of conditions implying that the Abel equation has a center may be interpreted as a simplified version of the classical Center-Focus problem of Poincaré. The Abel equation is said to have a "parametric center" if for each ε ∈ \mathbb R the equation y'=p(x)y2+ε q(x)y3 has a center. In this paper we show that the Abel equation has a parametric center if and only if the antiderivatives P=∫ p(x) dx, Q=∫ q(x) dx satisfy the equalities P=\widetilde P ∘ W, Q=\widetilde Q∘ W for some polynomials \widetilde P, \widetilde Q, and W such that W(a)=W(b). We also show that the last condition is necessary and sufficient for the "generalized moments" ∫ab Pid Q and ∫ab Qid P to vanish for all i≥ 0.