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

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.

Some properties of Melnikov functions near a cuspidal loop

In this paper,we consider the first-order Melnikov functions and limit cycle bifurcations of a nearHamiltonian system near a cuspidal loop.By establishing relations between the coefficients in the expansions of the two Melnikov functions,we give a general method to obtain the number of limit cycles near the cuspidal loop.As an application,we consider a kind of Liénard systems and obtain a new estimation on the lower bound of the maximum number of limit cycles.

On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.

Bifurcations of Invariant Tori and Subharmonic Solutions of Singularly Perturbed System

这篇论文处理为一个非自治的不可思议地使不安的系统在快动力学从限制周期产生的次谐波解决方案和 invarianttori 的分叉。把地图，一系列发作转变和积分的理论基于 Poincare 歧管，为不变的岩山 i 的存在的条件被获得，并且次谐波答案的 saddle-nodebifurcations 被学习。

Existence of Canards under Non-generic Conditions

The canard phenomenon occurring in planar fast-slow systems under non-generic conditions is investigated.When the critical manifold has a non-generic fold point,by using the method of asymptotic analysis combined with the recently developed blow-up technique,the existence of a canard is established and the asymptotic expansion of the parameter for which a canard exists is obtained.

Global Bifurcation of a Perturbed Double-Homoclinic Loop

这篇论文与不安处理一种第四个度系统。由使用多参数不安理论的方法；质的分析，系统能让六限制周期，这被证明。

Further study on Horozov-Iliev's method of estimating the number of limit cycles

In the study of the number of limit cycles of near-Hamiltonian systems,the first order Melnikov function plays an important role.This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function.