Real Algebraic and Analytic Geometry |

An ordered structure of rank two related to Dulac's Problem.

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homepage: http://www.math.toronto.edu/~speisseg/

Submission: 2006, March 2.

*Abstract:
For a vector field $\xi$ on $\RR^2$ 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 $\urank$-rank and the other involving the notion
of o-minimality.
.*

Mathematics Subject Classification (2000): 37C27, 03C64.

Keywords and Phrases: Vector fields, limit cycles, model theory, ordered structures.

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