Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"

Details Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles" Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"

2004-02-13

arxiv.org/abs/math/0402214v1

Proc. of the Steklov Inst. of Math., Vol. 254(1) / Sep., 2006

Mathematics

Dynamical Systems

23 pages, 5 figures

Scientific paper

10.1134/S0081543806030126

The paper deals with planar polynomial vector fields. We aim to estimate the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert's 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. This paper is the continuation of the author's paper in [Mosc. Math. J. 1 (2001), no. 4] where the lower bound of the form 2^{cn^2} has been obtained. Here we obtain the upper bound of the same form. We also associate an equipped planar graph to every planar polynomial vector field, this graph is a complete invariant for orbital topological classification of such fields.

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