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Bifurcation of limit cycles in a quintic system with ten parameters

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Abstract

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of the computer algebra system MATHEMATICA, the first 11 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 11 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth, we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems. The results of Jiang et al. (Int. J. Bifurcation Chaos 19:2107–2113, 2009) are improved.

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Acknowledgements

This research is partially supported by the National Nature Science Foundation of China (11201211, 61273012) and Nature Science Foundation of Shandong Province (ZR2012AL04).

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Authors and Affiliations

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

    Feng Li

  2. Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK

    Miao Wang

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  1. Feng Li
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  2. Miao Wang
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Correspondence to Feng Li.

Appendix

Appendix

Detailed recursive MATHEMATICA code to compute the quasi–Lyapunov constants at the origin of system (2):

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Li, F., Wang, M. Bifurcation of limit cycles in a quintic system with ten parameters. Nonlinear Dyn 71, 213–222 (2013). https://doi.org/10.1007/s11071-012-0653-2

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