Vanishing of higher-order moments on Lipschitz curves
This paper concerns the ordinary differential equation \[ dv / dx = \sum_{j=1}^\infty a_j(x) v^{j+1},\tag{*} \] where \(x \in [0, T]\) and the coefficients \(a_j\) lie in the space of bounded measurable complex-valued functions on \([0,T]\), equipped with the sup norm. This equation is said to determine a center provided every sufficiently small solution \(v\) satisfies \(v(0) = v(T)\). The well-known center problem for polynomial systems of ordinary equations on the plane at a singularity at which the eigenvalues are nonzero pure imaginary numbers can be fit into this framework. In earlier work, the author constructed an algebraic model for the center problem, a basic object of which is a topological group defined by the coefficients of the differential equation (*). The present paper is devoted to a study of that group. The author obtains a topological characterization of a Lipschitz curve defined by integrals of the coefficients of (*) with the property that all moments of order \(n \in {\mathbb N}\) vanish on it.
scientific article; zbMATH DE number 5275727
scientific article