Limit cycle enumeration in random vector fields

TL;DR

This paper investigates the expected number and distribution of limit cycles in random planar vector fields, providing bounds, limit laws, and asymptotics using advanced probabilistic and dynamical systems techniques.

Contribution

It introduces new bounds, limit laws, and asymptotic results for limit cycles in random polynomial vector fields, combining methods from dynamical systems and random analysis.

Findings

Lower bound on average number of limit cycles in Kostlan-Shub-Smale ensemble

Limit law for number of limit cycles near a perturbed center focus

Asymptotic counts for limit cycles under infinitesimal perturbations

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.