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Double canard cycles in singularly perturbed planar systems

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Abstract

We study the bifurcations of slow-fast cycles with two canard points in singularly perturbed planar systems. After analyzing the local dynamics of two canard points lying on the S-shaped critical manifolds, we give a sufficient condition under which there exist three hyperbolic limit cycles bifurcating from some slow-fast cycles. The proof is based on the geometric singular perturbation theory. Then, we apply the results to cubic Liénard equations with quadratic damping, and prove the coexistence of three large limit cycles enclosing three equilibria. This is a new dynamical configuration and has never been previously found in the existing references.

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Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

We would like to thank the editor and the anonymous referees for carefully reading the manuscript and providing valuable comments. This work was partly supported by the NSFC grants 11771449, 11771161, and the Science and Technology Project for Excellent Postdoctors of Hubei Province, China.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Sciences and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Shuang Chen & Ji Li

  2. Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Shuang Chen

  3. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Jinqiao Duan

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  1. Shuang Chen
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  2. Jinqiao Duan
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  3. Ji Li
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Correspondence to Shuang Chen.

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Chen, S., Duan, J. & Li, J. Double canard cycles in singularly perturbed planar systems. Nonlinear Dyn 105, 3715–3730 (2021). https://doi.org/10.1007/s11071-021-06769-6

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