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Existence and analytical approximations of limit cycles in a three-dimensional nonlinear autonomous feedback control system

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Abstract

This paper is concerned with the existence and the analytical approximations of limit cycles in a three-dimensional nonlinear autonomous feedback control system. Based on three-dimensional Hopf bifurcation theorem, the existence of limit cycles is first proved. Then the homotopy analysis method (HAM) is applied to obtain the analytical approximations of the limit cycle and its frequency. In deriving the higher-order approximations, the authors utilized the idea of a perturbation procedure proposed for limit cycles’ approximation in van der Pol equation. By comparing with the numerical integration solutions, it is shown that the accuracy of the analytical results obtained in this paper is very high, even when the amplitude of the limit cycle is large.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, China

    Huaxiong Chen, Jianhe Shen & Zheyan Zhou

Authors
  1. Huaxiong Chen
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  2. Jianhe Shen
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  3. Zheyan Zhou
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Corresponding author

Correspondence to Jianhe Shen.

Additional information

This research is supported by the National Natural Science Foundations of China under Grant Nos. 11201072 and 11102041, the China Postdoctoral Science Foundation under Grant No. 2011M500803, and Education Department of Fujian Province under Grant No. JA10065.

This paper was recommended for publication by Editor LÜ Jinhu.

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Chen, H., Shen, J. & Zhou, Z. Existence and analytical approximations of limit cycles in a three-dimensional nonlinear autonomous feedback control system. J Syst Sci Complex 27, 1158–1171 (2014). https://doi.org/10.1007/s11424-014-2015-2

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  • DOI: https://doi.org/10.1007/s11424-014-2015-2

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