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About Integrability of the Degenerate System

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Computer Algebra in Scientific Computing (CASC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11661))

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Abstract

We study integrability of an autonomous planar polynomial system of ODEs with a degenerate singular point at the origin depending on five parameters. By mean of the Power Geometry Method, this degenerated system is reduced to a non-degenerate form by the blow-up process. After, we search for the necessary conditions of local integrability by the normal form method. We look for the set of necessary conditions on parameters under which the original system is locally integrable near the degenerate stationary point. We found seven two-parametric families in the five-parameter space. Then first integrals of motion were found for six families. For the seventh family, we found the formal first integral. So, at least six of these families in parameters space are manifolds where the global integrability of the original system takes place.

The publication has been prepared with the support of the ”RUDN University Program 5–100”.

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Acknowledgements

The author is very grateful to Profs. A.D. Bruno, V.G. Romanovski, and A.B. Batkhin for important advices, discussions, and assistance.

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Authors and Affiliations

  1. Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Leninskie Gory 1(2), Moscow, 119991, Russian Federation

    Victor F. Edneral

  2. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya street, Moscow, 117198, Russian Federation

    Victor F. Edneral

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  1. Victor F. Edneral
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Correspondence to Victor F. Edneral .

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Editors and Affiliations

  1. Coventry University, Coventry, UK

    Matthew England

  2. University of Kassel, Kassel, Germany

    Wolfram Koepf

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. University of Kassel, Kassel, Germany

    Werner M. Seiler

  5. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Edneral, V.F. (2019). About Integrability of the Degenerate System. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_10

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  • DOI: https://doi.org/10.1007/978-3-030-26831-2_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26830-5

  • Online ISBN: 978-3-030-26831-2

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