An ordered structure of rank two related to Dulac's Problem
Volume 198 / 2008
Fundamenta Mathematicae 198 (2008), 17-60
MSC: 37C27, 03C64.
DOI: 10.4064/fm198-1-2
Abstract
For a vector field $\xi$ on $\mathbb{R}^2$ we construct, under certain assumptions on $\xi$, an ordered model-theoretic structure associated to the flow of $\xi$. We do this in such a way that the set of all limit cycles of $\xi$ is represented by a definable set. This allows us to give two restatements of Dulac's Problem for $\xi$—that is, the question whether $\xi$ has finitely many limit cycles—in model-theoretic terms, one involving the recently developed notion of ${\rm U}^{\rm l}\!\!\!\!\rm^{^o}$-rank and the other involving the notion of o-minimality.
