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Estimating Limit Cycle Bifurcations from Centers

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Differential Equations with Symbolic Computation

Part of the book series: Trends in Mathematics ((TM))

Abstract

We consider a simple computational approach to estimating the cyclicity of centers in various classes of planar polynomial systems. Among the results we establish are confirmation of Żoł¸dek’s result that at least 11 limit cycles can bifurcate from a cubic center, a quartic system with 17 limit cycles bifurcating from a non-degenerate center, and another quartic system with at least 22 limit cycles globally.

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References

  1. N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Amer. Math. Soc. Translation 100 (1954), 19 pp.

    Google Scholar 

  2. C. Christopher, An algebraic approach to the classification of centers in polynomial Liénard systems, J. Math. Anal. Appl. 229no. 1 (1999), 319–329.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Christopher and S. Lynch, Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces. Nonlinearity 12no. 4 (1999), 1099–1112.

    Article  MathSciNet  Google Scholar 

  4. C. Christopher and N. G. Lloyd, Polynomial systems: a lower bound for the Hilbert numbers. Proc. Roy. Soc. London Ser. A 450no. 1938 (1995), 219–224.

    MathSciNet  Google Scholar 

  5. G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 2.0.3: A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de.

    Google Scholar 

  6. E. M. James and N. G. Lloyd, A cubic system with eight small-amplitude limit cycles. IMA J. Appl. Math. 47no. 2 (1991), 163–171.

    MathSciNet  Google Scholar 

  7. Ji Bin Li and Qi Ming Huang, Bifurcations of limit cycles forming compound eyes in the cubic system. Chinese Ann. Math. Ser. B 8no. 4 (1987), 391–403.

    MathSciNet  Google Scholar 

  8. N. G. Lloyd, C. Christopher, J. Devlin, J. M. Pearson, and N. Yasmin, Quadratic-like cubic systems. Planar nonlinear dynamical systems (Delft, 1995). Differential Equations Dynam. Systems 5no. 3-4 (1997), 329–345.

    MathSciNet  Google Scholar 

  9. N. G. Lloyd and J. M. Pearson, Computing centre conditions for certain cubic systems. J. Comput. Appl. Math. 40no. 3 (1992), 323–336.

    Article  MathSciNet  Google Scholar 

  10. N. G. Lloyd and J. M. Pearson, Conditions for a centre and the bifurcation of limit cycles in a class of cubic systems. Bifurcations of planar vector fields (Luminy, 1989), Lecture Notes in Math. 1455, Springer, Berlin, 1990, 230–242.

    Google Scholar 

  11. K. S. Sibirskii, On the number of limit cycles arising from a singular point of focus or center type. Dokl. Akad. Nauk SSSR 161 (1965), 304–307 (Russian). Soviet Math. Dokl. 6 (1965), 428–431.

    MATH  MathSciNet  Google Scholar 

  12. J. Sokulski, The beginning of classification of Darboux integrals for cubic systems with center. Differential Integral Equations, to appear.

    Google Scholar 

  13. P. Yu and H. Maoan, Twelve limit cycles in a cubic order planar system with ℤ2-symmetry. Communications on Pure and Applied Analysis 3no. 3 (2004), 515–526.

    MathSciNet  Google Scholar 

  14. H. Żoł¸dek, The solution of the center-focus problem. Preprint, University of Warsaw, 1992.

    Google Scholar 

  15. H. Żoł¸dek, On a certain generalization of Bautin’s theorem. Nonlinearity 7no. 1 (1994), 273–279.

    Google Scholar 

  16. H. Żoł¸dek, The classification of reversible cubic systems with center. Topol. Methods Nonlinear Anal. 4no. 1 (1994), 79–136.

    Google Scholar 

  17. H. Żoł¸dek, Remarks on: “The classification of reversible cubic systems with center” Topol. Methods Nonlinear Anal. 8no. 2 (1996), 335–342.

    Google Scholar 

  18. H. Żoł¸dek, Eleven small limit cycles in a cubic vector field. Nonlinearity 8no. 5 (1995), 843–860.

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of Plymouth, Plymouth, PL4 8AA, UK

    Colin Christopher

Authors
  1. Colin Christopher
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Editor information

Editors and Affiliations

  1. School of Science, Beihang University, 37 Xueyuan Road, Beijing, 100083, China

    Dongming Wang  & Zhiming Zheng  & 

  2. Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie - CNRS, 8, rue due Capitaine Scott, 75015, Paris, France

    Dongming Wang

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Christopher, C. (2005). Estimating Limit Cycle Bifurcations from Centers. In: Wang, D., Zheng, Z. (eds) Differential Equations with Symbolic Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7429-2_2

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