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Applications of Symbolic Calculations and Polynomial Invariants to the Classification of Singularities of Differential Systems

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Computer Algebra in Scientific Computing (CASC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8136))

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Abstract

The goal of this paper is to present applications of symbolic calculations and polynomial invariants to the problem of classifying planar polynomial systems of differential equations. For these applications, we use some previously defined, and some new polynomial invariants. This is part of a much larger work by the authors together with J.C. Artés and J. Llibre which is in progress. We show here how polynomial invariants and their symbolic calculations are instrumental in obtaining the bifurcation diagram of the global configurations of singularities (finite and infinite), of quadratic differential systems having a unique simple finite singularity. This bifurcation diagram is given in the twelve-dimensional space of the coefficients of the systems, and the bifurcation points form an algebraic set. The classification of singularities is done using the notion of geometric equivalence relation of configurations of singularities, which is finer than the topological equivalence. The bifurcation diagram is expressed in terms of polynomial invariants. The results can, therefore, be applied to any family of quadratic systems, given in any normal form. Determining the configurations of singularities for any family of quadratic systems thus becomes a simple task using computer symbolic calculations.

This work was supported by NSERC. The second author is partially supported by FP7-PEOPLE-2012-IRSES-316338 and by the grant 12.839.08.05F from SCSTD of ASM.

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Author information

Authors and Affiliations

  1. Département de Mathématiques et de Statistiques, Université de Montréal, Canada

    Dana Schlomiuk

  2. Institute of Mathematics and Computer Science, Academy of Science of Moldova, Moldova

    Nicolae Vulpe

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  1. Dana Schlomiuk
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  2. Nicolae Vulpe
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Editor information

Editors and Affiliations

  1. Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Technische Universität München, 85748, Garching, Germany

    Ernst W. Mayr

  4. Khristianovich Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Schlomiuk, D., Vulpe, N. (2013). Applications of Symbolic Calculations and Polynomial Invariants to the Classification of Singularities of Differential Systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2013. Lecture Notes in Computer Science, vol 8136. Springer, Cham. https://doi.org/10.1007/978-3-319-02297-0_28

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  • DOI: https://doi.org/10.1007/978-3-319-02297-0_28

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02296-3

  • Online ISBN: 978-3-319-02297-0

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