Skip to main content
SpringerLink
Log in

Normal Forms of Two p: − q Resonant Polynomial Vector Fields

  • Conference paper
Computer Algebra in Scientific Computing (CASC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6885))

Included in the following conference series:

Abstract

We investigate a property of normal forms of p: − q resonant vector fields, which is related to isochronicity. The problem is reduced to studying polynomial ideals and their varieties which is performed using tools of computational algebra.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Algaba, A., Reyes, M.: Characterizing isochronous points and computing isochronous sections. J. Math. Anal. Appl. 355, 564–576 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amel’kin, V.V., Lukashevich, N.A., Sadovskii, A.P.: Nonlinear Oscillations in Second Order Systems. Belarusian State University, Minsk (1982) (in Russian)

    Google Scholar 

  3. Bardet, M., Boussaada, I., Chouikha, A.R., Strelcyn, J.-M.: Isochronicity conditions for some planar polynomial systems II. Bull. Sci. Math. 135, 230–249 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bibikov, Y.N.: Local Theory of Nonlinear Analytic Ordinary Differential Equations. Lecture Notes in Mathematics, vol. 702. Springer, New York (1979)

    MATH  Google Scholar 

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

    Google Scholar 

  6. Chavarriga, J., Sabatini, M.: A survey of isochronous centers. Qual. Theory Dyn. Syst. 1, 1–70 (1999)

    Google Scholar 

  7. Christopher, C., Mardešić, P., Rousseau, C.: Normalizable, integrable, and linearizable saddle points for complex quadratic systems in ℂ2. J. Dynam. Control Systems 9, 311–363 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Edneral, V.F.: On algorithm of the normal form building. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 134–142. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  9. Fronville, A., Sadovski, A.P., Zoladek, H.: Solution of the 1: − 2 resonant center problem in the quadratic case. Fund. Math. 157, 191–207 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomials. J. Symbolic Comput. 6, 146–167 (1988)

    MATH  Google Scholar 

  11. Giné, J., Grau, M.: Characterization of isochronous foci for planar analytic differential systems. Proc. Roy. Soc. Edinburgh Sect. A 135(5), 985–998 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Greuel, G.M., Pfister, G., Schönemann, H.: Singular 3.0 A computer algebra system for polynomial computations. Centre for Computer Algebra. University of Kaiserslautern (2005), http://www.singular.uni-kl.de

  13. Han, M., Romanovski, V.G.: Isochronicity and normal forms of polynomial systems of ODEs. Submitted to Journal of Symbolic Computation

    Google Scholar 

  14. Hu, Z., Romanovski, V.G., Shafer, D.S.: 1:-3 resonant centers on image with homogeneous cubic nonlinearities. Computers & Mathematics with Applications 56(8), 1927–1940 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kadyrsizova, Z., Romanovski, V.: Linearizability of 1:-3 resonant system with homogeneous cubic nonlinearities. In: International Symposium on Symbolic and Algebraic Computation (ISSAC 2008), Hagenberg, Austria, July 20-23, pp. 255–260. The Association for Computing Machinery, New York (2008)

    Google Scholar 

  16. Kukles, I.S., Piskunov, N.S.: On the isochronism of oscillations for conservative and nonconservative systems. Dokl. Akad. Nauk. SSSR 17(9), 467–470 (1937)

    Google Scholar 

  17. Pfister, G., Decker, W., Schönemann, H., Laplagne, S.: primdec.lib. A Singular 3.0 library for computing primary decomposition and radical of ideals (2005)

    Google Scholar 

  18. Romanovski, V.G., Shafer, D.S.: The Center and Cyclicity Problems: a Computational Algebra Approach. Birkhäuser Boston, Inc., Boston (2009)

    MATH  Google Scholar 

  19. Rudenok, A.E.: Strong isochronicity of a center and a focus of systems with homogeneous nonlinearities. Differ. Uravn. 45(2), 154–161 (2009) (in Russian); translation in Differ. Equ. 45(2), 159–167 (2009)

    Google Scholar 

  20. Wang, Q.L., Liu, Y.R.: Generalized isochronous centers for complex systems. Acta Math. Sin. (Engl. Ser.) 26(9), 1779–1792 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, P.S., Guy, M.J.T., Davenport, J.H.: P-adic reconstruction of rational numbers. ACM SIGSAM Bull. 16(2), 2–3 (1982)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Victor Edneral

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

    Valery G. Romanovski

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

    Valery G. Romanovski

Authors
  1. Victor Edneral
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valery G. Romanovski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies (LIT), Joint Institute for Nuclear Research (JINR), 141980, Dubna, Russia

    Vladimir P. Gerdt

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

    Wolfram Koepf

  3. Institut für Informatik, Lehrstuhl für Effiziente Algorithmen, Technische Universität München, Boltzmannstraße 3, 85748, Garching, Germany

    Ernst W. Mayr

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

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Edneral, V., Romanovski, V.G. (2011). Normal Forms of Two p: − q Resonant Polynomial Vector Fields. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23568-9_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-23568-9_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23567-2

  • Online ISBN: 978-3-642-23568-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation