Journal of Engineering Mathematics

, Volume 92, Issue 1, pp 185–202 | Cite as

Approximation of limit cycles in two-dimensional nonlinear systems near a Hopf bifurcation by canonical transformations

  • Jianhe Shen
  • Huaxiong Chen
  • Zheyan Zhou
  • Shuhui Chen


A perturbation procedure is proposed that can determine the limit cycles and their associated frequencies in general two-dimensional nonlinear systems. There are three key points in the procedure – the derivation of the canonical system, a well-chosen parameter transformation, and suitable solution expansions. We show the efficiency of the perturbation method applied to a classical Rosenzweig–MacArthur predator–prey model. The explicit and analytical approximations to the limit cycles and their associated frequencies are derived. The good agreement between the analytical approximations and the numerically generated results shows the high accuracy and efficiency of the presented method, especially when the bifurcation parameter is moderately near the critical value.


Limit cycle Perturbation procedure Two-dimensional system 



Financial support from the Natural Science Foundation of China (10972240, 11201072, and 11102041) and the Nonlinear Analysis Innovation Team (IRTL1206) funded by Fujian Normal University is gratefully acknowledged. The authors would like to thank the referees for their proposed improvements.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Jianhe Shen
    • 1
  • Huaxiong Chen
    • 1
  • Zheyan Zhou
    • 1
  • Shuhui Chen
    • 2
  1. 1.School of Mathematics and Computer ScienceFujian Normal UniversityFuzhouPeople’s Republic of China
  2. 2.Department of Applied Mechanics and EngineeringSun Yat-sen UniversityGuangzhouPeople’s Republic of China

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