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Center, limit cycles and isochronous center of a Z 4-equivariant quintic system

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Abstract

In this paper, we study the limit cycles bifurcations of four fine focuses in Z 4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z 4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z 4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.

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Authors and Affiliations

  1. Department of Hu’nan Shaoyang University, Shaoyang, 422000, P. R. China

    Chao Xiong Du

  2. School of Mathematics, Central South University, Changsha, 410083, P. R. China

    Chao Xiong Du & Yi Rong Liu

  3. Department of Information Science, Hu’nan Commercial University, Changsha, 410083, P. R. China

    Hei Long Mi

Authors
  1. Chao Xiong Du
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  2. Hei Long Mi
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  3. Yi Rong Liu
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Correspondence to Chao Xiong Du.

Additional information

Partially supported by National Natural Science Foundation of China (Grant No. 10771196) and the Research Fund of Hunan Provincial Education Department (Grant No. 09A082) and Hunan Provincial Natural Science Foundation (Grant No. 10JJ5046)

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Du, C.X., Mi, H.L. & Liu, Y.R. Center, limit cycles and isochronous center of a Z 4-equivariant quintic system. Acta. Math. Sin.-English Ser. 26, 1183–1196 (2010). https://doi.org/10.1007/s10114-010-7541-9

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  • DOI: https://doi.org/10.1007/s10114-010-7541-9

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