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On the existence and uniqueness of limit cycles for hybrid oscillators

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Abstract

In this paper, we first give criteria on the existence and uniqueness of limit cycles for \(\ddot{x}+f(x,\dot{x})\dot{x}+g(x)=0\), which is widely used to model the hybrid mechanical oscillators such as Van der Pol–Duffing–Rayleigh oscillator, Van der Pol–Rayleigh oscillator and so on. By applying the criteria, we completely prove the uniqueness of limit cycles for Van der Pol–Duffing–Rayleigh oscillator and an asymmetric and nonsmooth Van der Pol–Rayleigh oscillator. Moreover, we provide the location of the limit cycle when it exists.

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Acknowledgements

The first authors is supported by the National Natural Science Foundation of China (No. 12171485). The second author is supported by the National Natural Science Foundation of China (Nos. 12271353; 11931016), the Innovation Program of Shanghai Municipal Education Commission (No. 2021-01-07-00-02-E00087) and the Natural Science Foundation of Shanghai, China (No. 20ZR1428700).

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Authors and Affiliations

  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Hebai Chen

  2. School of Mathematical Sciences, CAM-Shanghai, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Dongmei **ao

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  1. Hebai Chen
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  2. Dongmei **ao
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Correspondence to Dongmei **ao.

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The work was partially supported by National Key R & D Program of China (No. 2022YFA1005900).

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Chen, H., **ao, D. On the existence and uniqueness of limit cycles for hybrid oscillators. Annali di Matematica 202, 2049–2071 (2023). https://doi.org/10.1007/s10231-023-01312-3

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