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On New Integrals of the Algaba-Gamero-Garcia System

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Computer Algebra in Scientific Computing (CASC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10490))

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Abstract

We study local integrability of a plane autonomous polynomial system of ODEs depending on five parameters with a degenerate singular point at the origin. The approach is based on making use of the Power Geometry Method and the computation of normal forms. We look for the complete set of necessary conditions on parameters of the system under which the system is locally integrable near the degenerate stationary point. We found earlier that the sets of parameters satisfying these conditions consist of four two-parameter subsets in the full five-parameter co-space. Now we consider the special subcase of the case \(b^2 = 2/3\) and separate subsubcases when additional first integrals can exist. Here we have found two such integrals.

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Acknowledgements

Victor F. Edneral was supported by the grant NSh-7989.2016.2 of the President of Russian Federation and by the Ministry of Education and Science of the Russian Federation (Agreement number 02 A03.21.0008), Valery G. Romanovski was supported by the Slovenian Research Agency (research core funding No. P1-0306).

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics of RAS, Miusskaya Sq. 4, Moscow, 125047, Russia

    Alexander D. Bruno

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

    Victor F. Edneral

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

    Victor F. Edneral

  4. Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova 17, SI-2000, Maribor, Slovenia

    Valery G. Romanovski

  5. CAMTP – Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000, Maribor, Slovenia

    Valery G. Romanovski

  6. Faculty of Natural Science and Mathematics, University of Maribor, Koroška cesta 160, SI-2000, Maribor, Slovenia

    Valery G. Romanovski

Authors
  1. Alexander D. Bruno
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  2. Victor F. Edneral
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  3. Valery G. Romanovski
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Corresponding author

Correspondence to Victor F. Edneral .

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Editors and Affiliations

  1. Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Bruno, A.D., Edneral, V.F., Romanovski, V.G. (2017). On New Integrals of the Algaba-Gamero-Garcia System. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2017. Lecture Notes in Computer Science(), vol 10490. Springer, Cham. https://doi.org/10.1007/978-3-319-66320-3_4

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  • DOI: https://doi.org/10.1007/978-3-319-66320-3_4

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