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Phase Portraits of a Class of Cubic Systems with an Ellipse and a Straight Line as Invariant Algebraic Curves

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Abstract

In this paper we classify the phase portraits in the Poincaré disc of a class of cubic polynomial differential systems having an invariant ellipse and an invariant straight line. We prove that such a class of cubic polynomial differential systems have exactly 43 topologically different phase portraits in the Poincaré disc. Also we obtain that the invariant ellipse in two of these phase portraits is a limit cycle.

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Acknowledgements

We thank to the reviewers their comments which allow us to improve the results of this paper. The first author is partially supported by Iranian Ministry of Science, Research and Technology and Iran’s National Elites Foundation. The second autor is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grant PID2019-104658GB-I00, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran

    Ali Bakhshalizadeh

  2. Department de Matematiques, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Catalonia, Spain

    Jaume Llibre

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  1. Ali Bakhshalizadeh
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  2. Jaume Llibre
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Correspondence to Ali Bakhshalizadeh.

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Bakhshalizadeh, A., Llibre, J. Phase Portraits of a Class of Cubic Systems with an Ellipse and a Straight Line as Invariant Algebraic Curves. Differ Equ Dyn Syst (2022). https://doi.org/10.1007/s12591-022-00597-9

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