For a vector field ξ on ℝ² we construct, under certain assumptions on ξ, an ordered model-theoretic structure associated to the flow of ξ. We do this in such a way that the set of all limit cycles of ξ is represented by a definable set. This allows us to give two restatements of Dulac’s Problem for ξ - that is, the question whether ξ has finitely many limit cycles-in model-theoretic terms, one involving the recently developed notion of -rank and the other involving the notion of o-minimality.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-1-2,
author = {A. Dolich and P. Speissegger},
title = {An ordered structure of rank two related to Dulac's Problem},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {17-60},
zbl = {1221.03027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-1-2}
}
A. Dolich; P. Speissegger. An ordered structure of rank two related to Dulac's Problem. Fundamenta Mathematicae, Tome 201 (2008) pp. 17-60. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-1-2/