Limit Cycle Bifurcations of a Planar Near-Integrable System with Two Small Parameters

Abstract

In this paper we consider a class of polynomial planar system with two small parameters, ε and λ, satisfying 0 < ε ≪ λ ≪ 1. The corresponding first order Melnikov function M₁ with respect to ε depends on λ so that it has an expansion of the form ${M_1}(h,\lambda ) = \sum\limits_{k = 0}^\infty {{M_{1k}}(h){\lambda ^k}}$. Assume that M_1k′ (h) is the first non-zero coefficient in the expansion. Then by estimating the number of zeros of M_1k′ (h), we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for 0 < ε ≪ λ ≪ 1, when k′ = 0 or 1. In addition, for each k ∈ ℕ, an upper bound of the maximal number of zeros of M_1k(h), taking into account their multiplicities, is presented.