  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.
