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On Integrability of a Planar ODE System Near a Degenerate Stationary Point

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Computer Algebra in Scientific Computing (CASC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5743))

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Abstract

We consider an autonomous system of ordinary differential equations, which is solved with respect to derivatives. To study local integrability of the system near a degenerate stationary point, we use an approach based on Power Geometry method and on the computation of the resonant normal form. For a planar 5-parametric example of such system, we found the complete set of necessary and sufficient conditions on parameters of the system for which the system is locally integrable near a degenerate stationary point.

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References

  1. Bruno, A.D., Edneral, V.F.: Algorithmic analysis of local integrability. Dokl. Akademii Nauk 424(3), 299–303 (2009) (in Russian); Doklady Mathem. 79(1), 48–52 (2009) (in English)

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  2. Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. Fizmatlit, Moscow (1998) (in Russian); Elsevier Science, Amsterdam (2000) (in English)

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  3. Bruno, A.D.: Local Methods in Nonlinear Differential Equations, Nauka, Moscow (1979) (in Russian); Springer, Berlin (1989) (in English)

    Google Scholar 

  4. Bruno, A.D.: Analytical form of differential equations (I,II). Trudy Moskov. Mat. Obsc. 25, 119–262 (1971); 26, 199–239 (1972) (in Russian); Trans. Moscow Math. Soc.  25, 131–288; 26, 199–239 (1972) (in English)

    MathSciNet  Google Scholar 

  5. Algaba, A., Gamero, E., Garcia, C.: The integrability problem for a class of planar systems. Nonlinearity 22, 395–420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Siegel, C.L.: Vorlesungen über Himmelsmechanik. Springer, Berlin (1956) (in German); Fizmatlit, Moscow (1959) (in Russian)

    Book  MATH  Google Scholar 

  7. Edneral, V.F.: On algorithm of the normal form building. In: Ganzha, et al. (eds.) Proc. CASC 2007. LNCS, vol. 4770, pp. 134–142. Springer, Heidelberg (2007)

    Google Scholar 

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Author information

Authors and Affiliations

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

    Alexander Bruno

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

    Victor Edneral

Authors
  1. Alexander Bruno
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  2. Victor Edneral
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Editor information

Editors and Affiliations

  1. Joint Institute for Nuclear Research, Laboratory of Information Technologies, Dubna, Russia

    Vladimir P. Gerdt

  2. Technische Universität München‘, Institut für Informatik, Garching, Germany

    Ernst W. Mayr

  3. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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© 2009 Springer-Verlag Berlin Heidelberg

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Bruno, A., Edneral, V. (2009). On Integrability of a Planar ODE System Near a Degenerate Stationary Point. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2009. Lecture Notes in Computer Science, vol 5743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04103-7_4

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  • DOI: https://doi.org/10.1007/978-3-642-04103-7_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04102-0

  • Online ISBN: 978-3-642-04103-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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