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On the Number of Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Point

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Abstract

In this paper we investigate the number of small amplitude limit cycles bifurcated from a class of planar piecewise smooth quadratic systems with focus-parabolic type critical point having exactly one switching line. By considering the Lyapunov constants of the system, we obtain some conditions under which (0, 0) is a center or a focus of order eleven. We prove that at least ten limit cycles can bifurcate from (0, 0), which is a new lower bound of the cyclicity of piecewise smooth quadratic systems with focus-parabolic type critical point having exactly one switching line. For four of the center conditions, we prove that at least six limit cycles can bifurcate from (0, 0) without destroying the singularity by higher order perturbations techniques, one more than that obtained by considering the linear parts of the perturbed Lyapunov constants. Finally, the center-focus problem for a subclass of the system is completely solved.

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Acknowledgements

The authors are very grateful to the associate editor and the anonymous referees for their careful reading and valuable suggestions, which have notably improved the quality of this paper.

Funding

This work is supported by National Natural Science Foundation of China under Grant number 11971019.

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

    Qing Zhang & Zhengdong Du

Authors
  1. Qing Zhang
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  2. Zhengdong Du
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Contributions

QZ: Investigation, Methodology, Validation, Data curation, Software, Writing—original draft. ZD: Conceptualization, Investigation, Methodology, Validation, Data curation, Writing—original draft, Writing—reviewing and editing.

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Correspondence to Zhengdong Du.

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Zhang, Q., Du, Z. On the Number of Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Point. Mediterr. J. Math. 20, 277 (2023). https://doi.org/10.1007/s00009-023-02480-5

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  • DOI: https://doi.org/10.1007/s00009-023-02480-5

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