The number of limit cycles bifurcating from a randomly perturbed center

TL;DR

This paper investigates the average number of limit cycles emerging from a perturbed linear center, linking the problem to the zeros of a specific class of random polynomials and providing asymptotic results.

Contribution

It connects bifurcation analysis of limit cycles to the zeros of generalized Kac polynomials, offering asymptotic formulas and insights into the bifurcation problem.

Findings

Asymptotic formulas for the average number of real zeros of the polynomial

Results on the bifurcation of limit cycles from a perturbed center

Order estimates for the mean number of real roots in the subcritical regime

Abstract

We consider the average number of limit cycles that bifurcate from a randomly perturbed linear center where the perturbation consists of random (bivariate) polynomials with independent coefficients. This problem reduces, by way of classical perturbation theory of the Poincar\'e first return map, to a problem on the real zeros of a random \emph{univariate} polynomial $\displaystyle f_n(x) = \sum_{m=0}^n c_m \xi_m x^m$ with independent coefficients $\xi_m$ having mean zero, variance 1 and $c_m \sim m^{-1/2}$. This polynomial belongs to the class of {\it generalized Kac polynomials} at the critical regime. We provide asymptotics for the average number of real zeros and answer the question on bifurcating limit cycles. Additionally, we provide the correct order of the mean number of real roots in the subcritical regime.