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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Limit cycle enumeration in random vector fields
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by Erik Lundberg
Trans. Amer. Math. Soc. 376 (2023), 5693-5730
DOI: https://doi.org/10.1090/tran/8936
Published electronically: May 17, 2023

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 [Ann. of Math. (2) 154 (2001), pp. 227–243] 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.
References
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Bibliographic Information
  • Erik Lundberg
  • Affiliation: Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431
  • MR Author ID: 819273
  • ORCID: 0000-0001-9623-6023
  • Email: elundber@fau.edu
  • Received by editor(s): June 9, 2022
  • Received by editor(s) in revised form: January 19, 2023
  • Published electronically: May 17, 2023
  • Additional Notes: The author acknowledges support from the Simons Foundation (grant 712397)
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 5693-5730
  • MSC (2020): Primary 34C07, 60G60
  • DOI: https://doi.org/10.1090/tran/8936
  • MathSciNet review: 4630757