Limit cycle enumeration in random vector fields

We study the number and distribution of limit cycles a planar vector field whose component functions are random polynomials. prove lower bound on average when polynomials sampled from Kostlan-Shub-Smale ensemble. Investigating problem introduced by Brudnyi [Ann. Math. (2) 154 (2001), pp. 227–243] we also consider special local setting counting near randomly perturbed center focus, perturbation has i.i.d. coefficients, law showing that situated within disk radius less than unity converges almost surely to real zeros logarithmically-correlated univariate power series. infinitesimal perturbations where obtain precise asymptotics global count for family models. The proofs these results use novel combinations techniques dynamical systems analytic functions.