| Melnikov函数在含有两个幂零尖点的双异宿环附近的展开式 |
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| 引用本文： | 苗稼乐,杨俊敏. Melnikov函数在含有两个幂零尖点的双异宿环附近的展开式[J]. 上海师范大学学报(自然科学版), 2020, 49(3): 295-310 |
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| 作者姓名： | 苗稼乐 杨俊敏 |
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| 作者单位： | 河北师范大学数学科学学院,河北石家庄050024;河北师范大学数学科学学院,河北石家庄050024 |
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| 基金项目： | The Natural Science Foundation of China (11971145); The Natural Science Foundation of Hebei Province(A2019205133) |
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| 摘 要： | 考虑未扰Liénard系统ẋ=y，ẏ=-g（x），其中deg g（x）=7，当该系统分别含有2，3，4和5个奇点时，给出了其所有的不同拓扑类型的相图，并给出了Melnikov函数在含有2个幂零尖点和1个双曲鞍点的双异宿环附近的展开式和得到极限环的条件.
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| 关 键 词： | 极限环 Liénard系统 近哈密顿系统 异宿环 Melnikov函数 |
| 收稿时间： | 2020-03-10 |
On the expansion of the Melnikov function near a double heteroclinic loop with two nilpotent cusps |
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| MIAO Jiale and YANG Junmin. On the expansion of the Melnikov function near a double heteroclinic loop with two nilpotent cusps[J]. Journal of Shanghai Normal University(Natural Sciences), 2020, 49(3): 295-310 |
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| Authors: | MIAO Jiale and YANG Junmin |
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| Affiliation: | School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, Hebei, China and School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, Hebei, China |
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| Abstract: | In this paper, we give all the different topological types of phase portrait for the unperturbed Liénard system ẋ=y, ẏ=-g(x) in the case that deg g(x)=7 and the system has 2, 3, 4 and 5 singular points, respectively. We then give the expansion of Melnikov function near a double heteroclinic loop with two nilpotent cusps and one hyperbolic saddle. We also give the conditions to obtain the limit cycles. |
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| Keywords: | limit cycle Liénard system near-Hamiltonian system heteroclinic loop Melnikov function |
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