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Special Cubic Perturbations of the Duffing Oscillator \(x''=x-x^3\) Near the Eight-Loop

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Abstract

We find an upper bound for the number of limit cycles, bifurcating from the eight-loop of the Duffing oscillator \(x''= x-x^{3}\) under the special cubic perturbation

$$\begin{aligned} x''= x-x^{3}+\lambda _{1}y+\lambda _{2}x^{2}+\lambda _{3}xy+\lambda _{4}x^{2}y. \end{aligned}$$

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Acknowledgements

Part of this paper was written while the second author was visiting the Mathematics Institute of Toulouse.

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Authors and Affiliations

  1. Mathematics Institute of Toulouse, UMR 5219, Toulouse University, 31062, Toulouse, France

    Lubomir Gavrilov

  2. Department of Mathematics Faculty of Sciences and Humanity Studies at Al-Aflaj, Prince Sattam bin Abdulaziz University, Al Kharj-Riyadh, Saudi Arabia

    Ameni Gargouri

  3. National School of Electronics and Telecommunications of Sfax, Tunis Road, El Ons City, 3018, Sfax, Tunisia

    Bassem Ben Hamed

Authors
  1. Lubomir Gavrilov
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  2. Ameni Gargouri
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  3. Bassem Ben Hamed
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Correspondence to Bassem Ben Hamed.

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Gavrilov, L., Gargouri, A. & Hamed, B.B. Special Cubic Perturbations of the Duffing Oscillator \(x''=x-x^3\) Near the Eight-Loop. Mediterr. J. Math. 18, 229 (2021). https://doi.org/10.1007/s00009-021-01868-5

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  • DOI: https://doi.org/10.1007/s00009-021-01868-5

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