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Third Order Melnikov Functions of a Cubic Center under Cubic Perturbations

Author

Listed:
  • Yanwei Liu

    (College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China)

  • Tonghua Zhang

    (Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC 3122, Australia)

  • Xia Liu

    (College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China)

Abstract

In this paper, cubic perturbations of the integral system ( 1 + x ) 2 d H where H = ( x 2 + y 2 ) / 2 are considered. Some useful formulae are deduced that can be used to compute the first three Melnikov functions associated with the perturbed system. By employing the properties of the ETC system and the expressions of the Melnikov functions, the existence of exactly six limit cycles is given. Note that there are many cases for the existence of third-order Melnikov functions, and some existence conditions are very complicated—the corresponding Melnikov functions are not presented.

Suggested Citation

  • Yanwei Liu & Tonghua Zhang & Xia Liu, 2022. "Third Order Melnikov Functions of a Cubic Center under Cubic Perturbations," Mathematics, MDPI, vol. 10(11), pages 1-17, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1816-:d:823803
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    References listed on IDEAS

    as
    1. Asheghi, R. & Nabavi, A., 2020. "The third order melnikov function of a cubic integrable system under quadratic perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
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