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The Perturbation of a Class of Hyper-Elliptic Hamilton System with a Double Eight Figure Loop

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Abstract

In this paper, we consider the upper bounds of the number of isolated zeros of Abelian integrals associated to system

$$\begin{aligned} \dot{x}=y, \dot{y}=-\left( x^5-\frac{5}{2}x^3+x\right) \end{aligned}$$

under the perturbation of \(\epsilon (\alpha _0+\alpha _1x+\alpha _2x^2+\alpha _3x^3+\alpha _4x^4)y\frac{\partial }{\partial y}\), where \(0<|\epsilon |\ll 1\) and \(\alpha _i\in \mathbb {R}, i=0,1,2,3,4\). The unperturbed system has a double eight figure loop. The sharp upper bounds are obtained for ten cases that two of five parameters vanish.

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Acknowledgments

This work was supported by the National Science Foundation of China (11271046, 11361046), the Science and Technology Pillar Program of Ningxia (KJ[2015]26(4)), the Fundamental Research Funds for the Central Universities, the Natural Science Foundation of Ningxia (NZ13213), the Higher Education Science Program of Ningxia (GX2014[222]17) and the key Program of Higher Education of Henan (16A110038).

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Ningxia Normal University, Guyuan, 756000, People’s Republic of China

    Jihua Yang

  2. Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Jihua Yang & Liqin Zhao

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  1. Jihua Yang
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  2. Liqin Zhao
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Correspondence to Liqin Zhao.

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Yang, J., Zhao, L. The Perturbation of a Class of Hyper-Elliptic Hamilton System with a Double Eight Figure Loop. Qual. Theory Dyn. Syst. 16, 317–360 (2017). https://doi.org/10.1007/s12346-016-0194-z

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  • DOI: https://doi.org/10.1007/s12346-016-0194-z

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