Limit cycle enumeration in random vector fields
Abstract
We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from the Kostlan-Shub-Smale ensemble. Investigating a problem introduced by Brudnyi [Annals of Mathematics (2001)] we also consider a special local setting of counting limit cycles near a randomly perturbed center focus, and when the perturbation has i.i.d. coefficients, we prove a limit law showing that the number of limit cycles situated within a disk of radius less than unity converges almost surely to the number of real zeros of a logarithmically-correlated random univariate power series. We also consider infinitesimal perturbations where we obtain precise asymptotics on the global average count of limit cycles for a family of models. The proofs of these results use novel combinations of techniques from dynamical systems and random analytic functions.
Keywords
Cite
@article{arxiv.2007.00724,
title = {Limit cycle enumeration in random vector fields},
author = {Erik Lundberg},
journal= {arXiv preprint arXiv:2007.00724},
year = {2023}
}
Comments
41 pages. In addition to the updates in version 2, this version includes several minor revisions and a sketch of an alternate proof of Theorem 3 using results of A. Lerario and M. Stecconi. The open problem concerning infinitesimal perturbations has been resolved (see the comment in the introduction "added in press"). The paper will appear in Transactions of the American Mathematical Society