A trichotomy for expansions of Ran by trajectories of analytic planar vector fields, Unpublished note, available at http://www.math.ohiostate.edu/∼miller/trichot.pdf

Caveat. This note has become seriously out of date due, among other things, to recent work by Philipp Hieronymi [Defining the set of integers in expansions of the real field by a closed discrete set. Proc. Amer. Math. Soc. 138 (2010), no. 6, 21632168. MR2596055]. In particular, the paragraph after the proof of Corollary 4 should be struck. Rather than update this particular note, we are are actively working on an actual submittable paper. See also [Miller, Chris Expansions of o-minimal structures on the real field by trajectories of linear vector fields. Proc. Amer. Math. Soc. 139 (2011), no. 1, 319330. MR2729094] for related current developments. We are interested in expanding o-minimal structures on the real field by trajectories1 of definable vector fields. This note2 is a preliminary report on some progress. We do not attempt to state and prove results as efficiently as possible or in the greatest generality. The reader is assumed to be familiar with o-minimal expansions of the real field; see [4–6] for basic references and surveys. We also need a fair amount of ODE theory, some quite basic, some less so; see e.g. [14].3 An important source of inspiration was the essentially expository [1, Ch. 5]. See [8, 9, 12] for some related material, and [11] for discussion of