LIMIT CYCLE BIFURCATIONS OF A PLANARNEAR-INTEGRABLE SYSTEM WITH TWO SMALL PARAMETERS

摘要

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 M_(1) with respect to ε depends on λ so that it has an expansion of the form M_(1)(h,λ)=∑k=0∞M_(1k)(h)λ^(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.