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Complete study on a bi-center problem for the Z 2-equivariant cubic vector fields

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Abstract

For the planar Z 2-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Liapunov constants are completely discussed. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z 2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme 〈6 ∐ 6〉 is proved.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, P. R. China

    Yi Rong Liu & Ji Bin Li

  2. School of Science, Kunming University of Science and Technology, Kunming, 650093, P. R. China

    Ji Bin Li

Authors
  1. Yi Rong Liu
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  2. Ji Bin Li
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Correspondence to Yi Rong Liu.

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Supported by National Natural Science Foundation of China (Grant Nos. 10831003 and 10771196)

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Liu, Y.R., Li, J.B. Complete study on a bi-center problem for the Z 2-equivariant cubic vector fields. Acta. Math. Sin.-English Ser. 27, 1379–1394 (2011). https://doi.org/10.1007/s10114-011-8412-8

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  • DOI: https://doi.org/10.1007/s10114-011-8412-8

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