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Perturbation of a Period Annulus Bounded by a Saddle–Saddle Cycle in a Hyperelliptic Hamiltonian Systems of Degree Seven

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Abstract

In this paper, we study the limit cycle bifurcation by perturbing the period annuluses of two perturbed hyper-elliptic Hamiltonian systems of degree seven. The period annuluses are bounded by heteroclinic loops, inside or outside of which there exist two nilpotent cusps. The bifurcation function is Abelian integral which is the first-order approximation of the Poincaré map. The sharp bounds of the number of limit cycles bifurcated from the periodic annuluses are obtained by Chebyshev criterion and asymptotic analysis.

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Funding

This work was jointly supported by Program for Innovative Team of GUFE (2018-2021), and Te Project of Young Teacher’s Upgrading Program in Guangxi Province (2019KY1310), National Natural Science Foundation of China (Grant No. 11671176) and Natural Science Foundation of Zhejiang Province under Grant (No. LY20A010016).

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Authors and Affiliations

  1. Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning, 530003, Guangxi, People’s Republic of China

    Sumin Yang, Bin Qin & Guoen Xia

  2. Department of Human Sciences, Guangxi Technological College of Machinery and Electricity, Nanning, 530007, Guangxi, China

    Sumin Yang

  3. School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321000, People’s Republic of China

    Yong-Hui Xia

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  1. Sumin Yang
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  2. Bin Qin
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  3. Guoen Xia
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  4. Yong-Hui Xia
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Correspondence to Yong-Hui Xia.

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Yang, S., Qin, B., Xia, G. et al. Perturbation of a Period Annulus Bounded by a Saddle–Saddle Cycle in a Hyperelliptic Hamiltonian Systems of Degree Seven. Qual. Theory Dyn. Syst. 19, 33 (2020). https://doi.org/10.1007/s12346-020-00348-7

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