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Anti-control of Hopf bifurcation in the Shimizu–Morioka system using an explicit criterion

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Abstract

We consider anti-control of Hopf bifurcation for the Shimizu–Morioka system by using an explicit criterion. We first provide the two conditions for the existence of Hopf bifurcation, that is, eigenvalue assignment and transversality conditions, which could be formulated through the coefficients of characteristic equation, and the obtained conditions do not need to calculate the eigenvalue and eigenvalue’s derivatives. The center manifold theory and normal form reduction are utilized to derive the nonlinear gains for controlling the stability of the created limit circle. In addition, we further improve the computing formulas of amplitude and frequency of Hopf limit cycle. Numerical analysis also verifies the effectiveness of the proposed results.

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Acknowledgements

This work was supported in part by the National Key Research and Development Program of China under Grant 2016 YFB0800601, in part by the National Natural Science Foundation of China under Grant 61472331, in part by the Talents of Science and Technology Promote Plan, Chongqing Science & Technology Commission, in part by the National Natural Science Foundation of Hubei province of China (2015CFB264), in part by the National Natural Science Foundation of China under Grant 61503310.

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Authors and Affiliations

  1. The College of Electronic and Information Engineering, Southwest University, Chongqing, 400715, People’s Republic of China

    Yi Yang, Xiaofeng Liao & Tao Dong

  2. The Science and Technology College, Hubei University For Nationalities, Enshi, 445000, People’s Republic of China

    Yi Yang

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  1. Yi Yang
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  2. Xiaofeng Liao
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  3. Tao Dong
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Correspondence to Xiaofeng Liao.

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Yang, Y., Liao, X. & Dong, T. Anti-control of Hopf bifurcation in the Shimizu–Morioka system using an explicit criterion. Nonlinear Dyn 89, 1453–1461 (2017). https://doi.org/10.1007/s11071-017-3527-9

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  • DOI: https://doi.org/10.1007/s11071-017-3527-9

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