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Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

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Abstract

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters µ and λ with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.

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Authors and Affiliations

  1. Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou, 510275, P. R. China

    Cheng-biao Huang  (黄赪彪) & Jia Liu  (刘佳)

Authors
  1. Cheng-biao Huang  (黄赪彪)
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  2. Jia Liu  (刘佳)
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Corresponding author

Correspondence to Cheng-biao Huang  (黄赪彪).

Additional information

Communicated by WANG Biao

Project supported by the National Natural Science Foundation of China (No. 10672193)

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Huang, Cb., Liu, J. Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system. Appl. Math. Mech.-Engl. Ed. 29, 1195–1201 (2008). https://doi.org/10.1007/s10483-008-0908-6

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  • DOI: https://doi.org/10.1007/s10483-008-0908-6

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2000 Mathematics Subject Classification

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