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The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems

Author

Listed:
  • Lijun Wei

    (School of Mathematics, Hangzhou Normal University, Hangzhou 310036, China)

  • Yun Tian

    (Department of Mathematics, Shanghai Normal University, Shanghai 200234, China)

  • Yancong Xu

    (School of Mathematics, Hangzhou Normal University, Hangzhou 310036, China)

Abstract

This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function H = 1 2 y 2 + 1 2 x 2 − 2 3 x 3 + a 4 x 4 ( a ≠ 0 ) under two types of polynomial perturbations of degree m , respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least [ 3 m − 1 4 ] small limit cycles near the center, where m ≤ 101 , and that the related near-Hamiltonian system with general polynomial perturbations can have at least m + [ m + 1 2 ] − 2 small-amplitude limit cycles, where m ≤ 16 . Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, [ · ] represents the integer function.

Suggested Citation

  • Lijun Wei & Yun Tian & Yancong Xu, 2022. "The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems," Mathematics, MDPI, vol. 10(9), pages 1-14, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1483-:d:805378
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    References listed on IDEAS

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    1. Barbara Arcet & Valery G. Romanovski, 2021. "On Some Reversible Cubic Systems," Mathematics, MDPI, vol. 9(12), pages 1-20, June.
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