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Bifurcations of limit circles and center conditions for a class of non-analytic cubic Z 2 polynomial differential systems

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Abstract

In this paper, bifurcations of limit cycles at three fine focuses for a class of Z 2-equivariant non-analytic cubic planar differential systems are studied. By a transformation, we first transform nonanalytic systems into analytic systems. Then sufficient and necessary conditions for critical points of the systems being centers are obtained. The fact that there exist 12 small amplitude limit cycles created from the critical points is also proved. Henceforth we give a lower bound of cyclicity of Z 2-equivariant non-analytic cubic differential systems.

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References

  1. Li, J.: Hilberts 16th problem and bifurcation of Planar polynomial vector fields. Int. J. Bifurcation Chaos, 13, 47–106 (2003)

    Article  MATH  Google Scholar 

  2. Yu, P., Han, M. A.: Small limit cycles bifurcation from fine focus points in cubic order Z 2-equivariant vector fields. Chaos, Solitons and Fractals, 24, 329–348 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Lloyd, N. G., Pearson J. M.: Reduce and the bifurcation of limit cycles. J. Symbolic Comput., 9, 215–224 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Liu, Y., Li, J.: Theory of values of singular point in complex autonomous differential system. Sci. China Ser. A, 3, 245–255 (1989)

    Google Scholar 

  5. Liu, Y.: Theory of center-focus for a class of higher-degree critical points and infinite points. Sci. China Ser. A, 44, 37–48 (2001)

    Google Scholar 

  6. Shi, S. L.: A concrete example of the existence of four limit cycles for quadratic systems. Sci. China Ser. A, 23, 16–21 (1980)

    Google Scholar 

  7. Chen, L. S., Wang, M. S.: The relative position and number of limit cycles of a quadratic differential system. Acta Mathematica Sinica, Chinese Series, 22(6), 751–758 (1979)

    MATH  Google Scholar 

  8. Liu, Y., Huang, W. T.: A cubic system with twelve small amplitude limit cycles. Bull. Sci. Math., 129, 83–98 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, Q. L., Liu, Y. R., Du, C. X.: Small limit cycles bifurcating from fine focus points in quartic order Z 3-equivariant vector fields. J. Math. Anal. Appl., 337, 524–536 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wu, Y. H., Gao, Y. X., Han, M. A.: Bifurcations of the limit cycles in a Z 3-equivariant quartic planar vector field. Chaos, Solitons and Fractals, 38, 1177–1186 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yu, P., Han, M., Yuan, Y.: Analysis on limit cycles of Z q-equivariant polynomial vector fields with degree 3 or 4. J. Math. Anal. Appl., 322, 51–65 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Du, C., Mi, H., Liu, Y.: Center, limit cycles and isochronous center of a Z 4-equivariant quintic system. Acta Mathematica Sinica, English Series, 26(6), 1183–1191 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Llibre, J., Valls, C.: Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities. J. Differential Equations, 246, 2192–2204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Llibre, J., Valls, C.: Classification of the centers and isochronous centers for a class of quartic-like systems. Nonlinear Analysis, 71, 3119–3128 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Llibre, J., Valls, C.: Classification of the centers, their cyclicity and isochronicity for the generalized quadratic polynomial differential systems. J. Math. Anal. Appl., 357, 427–437 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Llibre, J., Valls, C.: Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems. Nonlinear Differ. Equ. Appl., 16, 657–679 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, J.: The generalized focal values and bifurcations of limit cycles for quasi-quadratic system. Acta Mathematica Sinca, Chinese Series, 45, 671–682 (2002)

    Article  Google Scholar 

  18. Liu, Y., Li, J., Huang, W.: Singular point vaules, center problem and bifurcations of limit cycles of two dimensional differential autonomous systems. Science Press, China, 2009, 162–190

    Google Scholar 

  19. Liu, Y.: The Generalized focal values and bifurcation of limits cycles for quasi quadratic systems. Acta Mathematica Sinica, Chinese Series, 45, 671–682 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Xiao, P.: Critical point quantities and integrability conditions for complex planar resonant polynomial differential systems, PhD thesis, Central South University, 2005

  21. Liu, Y., Li, J.: Some Classical Problems about Planar Vector Fileds (in Chinese), Science Press (China), Beijing, 2010

    Google Scholar 

  22. Liu, Y., Chen, H.: Formulas of singular point quantities and the first 10 saddle quantities for a class of cubic system. Acta Math. Appl. Sinica, 25, 295–302 (2002)

    MATH  Google Scholar 

  23. Amelbkin, B. B., Lukasevnky, H. A., Catovcki, A. N.: Nonlinear Vibration, VGU Lenin Publ., Moscow, 1982, 19–21

    Google Scholar 

  24. Han, M. A., Lin, Y. P.: A study on the existence of limit cycles of a planar system with 3rd-degree polynomials. Int. J. Bifurcation Chaos, 14, 41–60 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Science, Linyi University, Linyi, 276005, P. R. China

    Feng Li & Yin Lai Jin

  2. School of Mathematical Science and Computing Technology, Central South University, Changsha, 410004, P. R. China

    Yi Rong Liu

Authors
  1. Feng Li
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  2. Yi Rong Liu
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  3. Yin Lai Jin
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Correspondence to Feng Li.

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Supported by National Nature Science Foundation of China (Grant Nos. 11071222, 11101126)

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Li, F., Liu, Y.R. & Jin, Y.L. Bifurcations of limit circles and center conditions for a class of non-analytic cubic Z 2 polynomial differential systems. Acta. Math. Sin.-English Ser. 28, 2275–2288 (2012). https://doi.org/10.1007/s10114-012-0454-z

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