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Non-existence and uniqueness of limit cycles in a class of generalized Liénard equations

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Abstract

We provide a sharp upper bound for the number of limit cycles of the generalized Liénard differential systems

$$\begin{aligned} \dot{x} =y+a x^n + b x^k, \quad \dot{y} =cx^m \end{aligned}$$

where n, k, m are positive integers, \(1<n < k \) and \(a,b,c \in \mathbb {R}\) with \(bc \ne 0\). We also provide examples realizing the upper bounds

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Acknowledgements

The first author is supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049–001, Lisboa, Portugal

    Claudia Valls

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  1. Jaume Llibre
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  2. Claudia Valls
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Correspondence to Claudia Valls.

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Llibre, J., Valls, C. Non-existence and uniqueness of limit cycles in a class of generalized Liénard equations. Bol. Soc. Mat. Mex. 28, 41 (2022). https://doi.org/10.1007/s40590-022-00433-8

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  • DOI: https://doi.org/10.1007/s40590-022-00433-8

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