Medium amplitude limit cycles in second order perturbed polynomial Liénard systems (Q1733808)

The paper is devoted to polynomial generalized Liénard systems as a result of second order polynomial perturbations of a Hamiltonian system with a linear center. The main purpose is to derive an upper bound for medium (amplitude) limit cycles of this system that is, limit cycles that bifurcate from some closed orbits of the Hamiltonian system. The authors find a generic (open and dense) subset in the space of perturbations of degree \(2l\), with \(l\geq1\), such that each associated perturbed generalized Liénard system has at most \(2l-1\) medium limit cycles. They show that this upper bound is reached and finally present some concrete examples. The proof of this result is based on the study of higher order Poincaré-Pontryagin-Melnikov functions of the displacement function associated with the Liénard system.