Qualitative Theory of Dynamical Systems

, Volume 9, Issue 1–2, pp 319–324

Focal Values of Plane Cubic Centers

Open Access
Article

Abstract

We prove that the vanishing of 11 focal values is not sufficient to ensure that a complex plane cubic system has a complex center. This is done by finding a complex cubic system with a high order weak focus using an extensive computer search.

References

  1. 1.
    Christopher, C.: Estimating limit cycle bifurcations from centers. In Differential Equations With Symbolic Computation, Trends Math., pp. 23–35. Birkhäuser, Basel (2005)Google Scholar
  2. 2.
    Frommer M.: Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmtheitsstellen. Math. Ann. 109, 395–424 (1934)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Graf v. Bothmer H.-C.: Cord Erdenberger, and Katharina Ludwig. A new family of rational surfaces in \({\mathbb{P}^{4}}\). J. Symbolic Comput. 39(1), 51–60 (2005)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Hearn, A.: REDUCE. Available at http://www.reduce-algebra.com (2004)
  5. 5.
    Schreyer, F.-O.: Small fields in constructive algebraic geometry. In Moduli of vector bundles (Sanda, 1994; Kyoto, 1994). Lecture Notes in Pure and Appl. Math., vol. 179, pp. 221–228. Dekker, New York (1996)Google Scholar
  6. 6.
    Graf v. Bothmer, H.-C., Martin, C.: A C++ program for calculating focal values in characteristic p. Available at http://www-ifm.math.uni-hannover.de/~bothmer/strudel (2005)
  7. 7.
    Graf v. Bothmer H.-C., Martin C.: Frommers algorithm. NoDEA 14(5–6), 694–698 (2007)Google Scholar
  8. 8.
    Graf v. Bothmer, H.-C., Kröker, J.: A improved C++ program for calculating focal values in characteristic p. Available at http://www.stud.uni-hannover.de/~kroeker/centerfocus/index.html or at http://sourceforge.net/projects/centerfocus/ (2009)
  9. 9.
    Graf v. Bothmer H.-C., Schreyer F.-O.: A quick and dirty irreducibility test for multivariate polynomials over \({{\mathbb{F}}_q}\). Experiment. Math. 14(4), 415–422 (2005)MathSciNetGoogle Scholar
  10. 10.
    Wolf v. Wahl.: Personal communication (2005)Google Scholar
  11. 11.
    Żoła̧dek H.: Eleven small limit cycles in a cubic vector field. Nonlinearity 8(5), 843–860 (1995)CrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Courant Research Centre “Higher Order Structures”, Mathematisches InstitiutGeorg-August-Universität GöttingenGöttingenGermany

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