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Puiseux Integrability of Differential Equations

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Abstract

In this work we study polynomial differential systems in the plane and define a new type of integrability that we call Puiseux integrability. As its name indicates, the Puiseux integrability is based on finding and studying the structure of Puiseux series that are solutions of a first order ordinary differential equation related to the original differential system. The necessary and sufficient conditions to have such integrability are given. These conditions are used to solve the integrability problem for quintic Liénard differential systems with a cubic damping function.

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Acknowledgements

The first author is partially supported by Russian Science Foundation Grant No. 19-71-10003. The second author is partially supported by the Agencia Estatal de Investigación grant PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276. The third author is partially supported by FCT/Portugal through projects UIDB/04459/2020 and UIDP/04459/2020.

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Authors and Affiliations

  1. Department of Applied Mathematics, HSE University, 34 Tallinskaya Street, 123458, Moscow, Russian Federation

    Maria V. Demina

  2. Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001, Lleida, Catalonia, Spain

    Jaume Giné

  3. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Claudia Valls

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  1. Maria V. Demina
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  2. Jaume Giné
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  3. Claudia Valls
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authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

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Correspondence to Maria V. Demina, Jaume Giné or Claudia Valls.

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Demina, M.V., Giné, J. & Valls, C. Puiseux Integrability of Differential Equations. Qual. Theory Dyn. Syst. 21, 35 (2022). https://doi.org/10.1007/s12346-022-00565-2

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