| Linear estimation and distribution of the limit cycles bifurcated from a kind of degenerate polycycles |
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| 引用本文： | Li-qin ZHAO Department of Mathematics,Beijing Normal University,Beijing 100875,China. Linear estimation and distribution of the limit cycles bifurcated from a kind of degenerate polycycles[J]. 中国科学A辑(英文版), 2007, 50(3): 334-344. DOI: 10.1007/s11425-007-0015-6 |
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| 作者姓名： | Li-qin ZHAO Department of Mathematics Beijing Normal University Beijing 100875 China |
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| 作者单位： | Li-qin ZHAO Department of Mathematics，Beijing Normal University，Beijing 100875，China |
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| 摘 要： | In this paper, we investigate the number and the distribution of the limit cycles bifurcated from a kind of degenerate planar polycycles through three singular points: a saddle-node P0, a fine saddle P1 with finite order m∈N, a contractive (attracting) saddle P2 with the hyperbolicity ratio q2(0)■Q. The connection between P0 and P1 is of hh-type and the connection between P0 and P2 is of hp-type. It is assumed that the connections between P0 to P2 and P0 to P1 keep unbroken. We obtain that the cyclicity of this polycycle is not more than 3m 1, which is linearly dependent on the order of the resonant saddle P1 We also show that the cyclicity is not more than m 3 if q2(0)>m, and that the nearer q2(0)is close to 1, the more the limit cycles are bifurcated.
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| 收稿时间： | 2005-05-17 |
| 修稿时间： | 2006-11-10 |
Linear estimation and distribution of the limit cycles bifurcated from a kind of degenerate polycycles |
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| Li-qin Zhao. Linear estimation and distribution of the limit cycles bifurcated from a kind of degenerate polycycles[J]. Science in China(Mathematics), 2007, 50(3): 334-344. DOI: 10.1007/s11425-007-0015-6 |
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| Authors: | Li-qin Zhao |
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| Affiliation: | Department of Mathematics, Beijing Normal University, Beijing 100875, China |
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| Abstract: | In this paper, we investigate the number and the distribution of the limit cycles bifurcated from a kind of degenerate planar polycycles through three singular points: a saddle-node P 0, a fine saddle P 1 with finite order m ∈ N, a contractive (attracting) saddle P 2 with the hyperbolicity ratio q 2(0) ∉ Q. The connection between P 0 and P 1 is of hh-type and the connection between P 0 and P 2 is of hp-type. It is assumed that the connections between P 0 to P 2 and P 0 to P 1 keep unbroken. We obtain that the cyclicity of this polycycle is not more than 3m + 1, which is linearly dependent on the order of the resonant saddle P 1. We also show that the cyclicity is not more than m + 3 if q 2(0) > m, and that the nearer q 2(0) is close to 1, the more the limit cycles are bifurcated. This work was supported by the National Natural Science Foundation of China (Grant No. 10671020) |
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| Keywords: | degenerate polycycle cyclicity finitely-smooth normal form transition map |
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