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Routes to bursting in active control system with multiple time delays

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Abstract

This paper investigates the generation of some novel bursting patterns in active control oscillator with multiple time delays. We present the bursting patterns, including symmetric codimension one and codimension two bursters with the slow variation of periodic excitation item. We calculate the bifurcation conditions of fast subsystem as well as its stability related to the time delay. We also identify some regimes of bursting depending on the magnitude of the delay itself and the strength of time delayed coupling in the model. Our results show that the dynamics of bursters in delayed system are quite different from those in systems without any delay. In particular, delay can be used as a tuning parameter to modulate dynamics of bursting corresponding to the different type. Furthermore, we use transformed phase space analysis to explore the evolution details of the delayed bursting behavior. Time delay can enhance the spiking performance and obtain the remarkable spiking dynamics even in a very simple model, which enriches the routes to bursting dynamics. Also some numerical simulations are included to illustrate the validity of our study.

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Acknowledgements

The authors thank the editor and anonymous reviewers for their valuable comments and suggestions that helped to improve the paper. The authors are supported by the National Natural Science Foundation of China (Grant Nos. 11472116, 11502091 and 11572141).

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Authors and Affiliations

  1. School of Science, Nantong University, Nantong, 226019, People’s Republic of China

    Yue Yu

  2. School of Mathematical Science, Huaiyin Normal University, Huaiyin, 223300, People’s Republic of China

    Chun Zhang

  3. Faculty of Science, Jiangsu University, Zhenjiang, 212013, People’s Republic of China

    Xiujing Han

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  1. Yue Yu
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  2. Chun Zhang
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  3. Xiujing Han
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Correspondence to Yue Yu.

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Yu, Y., Zhang, C. & Han, X. Routes to bursting in active control system with multiple time delays. Nonlinear Dyn 88, 2241–2254 (2017). https://doi.org/10.1007/s11071-017-3373-9

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