带同宿环的分段线性哈密顿系统的极限环分支
Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop
摘要：
研究了一类非光滑多项式微分系统的极限环分支，通过扰动一个以原点为中心，原点周围有一个同宿环的分段线性哈密顿系统，得到了系统的极限环分支.利用分段近哈密顿系统的第一Melnikov函数，给出了系统在Hopf分支和同宿分支中最大极限环个数的下界，并导出了系统在中心和同宿环之间的周期环上分支到一阶的极限环个数的上界.在扰动项次数较小时，我们得到了第一Melnikov函数零点个数的精确结果。
英文摘要：
In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclnic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.
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