An ordered structure of rank two related to Dulac's problem

Abstract

For a vector field F on the Euclidean plane we construct, under certain assumptions on F, an ordered model-theoretic structure associated to the flow of F. We do this in such a way that the set of all limit cycles of F is represented by a definable set. This allows us to give two restatements of Dulac's Problem for F--that is, the question whether F has finitely many limit cycles--in model-theoretic terms, one involving the recently developed notion of thorn-rank and the other involving the notion of o-minimality.