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Delayed Transcritical Bifurcation Induced Mixed Bursting in a Modified SM System with Asymmetrically Distributed Equilibria

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Abstract

Delayed bifurcation behavior is ubiquitous in multi-time scale systems and has a great important effect on bursting oscillations. In this paper, a parametrically driven Shimizu-Morioka system is proposed and distinct delay behavior is observed when the slow-varying parameter passes through the transcritical bifurcation point periodically. Periodic and chaotic bursting oscillations induced by such delay behavior are investigated by using the fast-slow analysis method with different excitation amplitudes. More interesting, the fast subsystem has a zero equilibrium point branch and two asymmetrically distributed equilibrium point branches with respect to the slow-varying parameter. Thus, there exist two possible routes for the trajectory to choose when the delayed transcritical bifurcation takes place. This leads to that the bursting patterns corresponding to different excitation periods may be different. As a result, two possible mixed bursting patterns, named as “delayed transcritical/transcritical” bursting of point-point type mixed with “delayed transcritical/supHopf/fold” bursting of point-cycle type and “delayed transcritical/subHopf/subHopf/fold” bursting of point-cycle type mixed with “delayed transcritical/supHopf/fold” bursting of point-cycle, are revealed. Furthermore, the effect of the excitation frequency on the delay behavior is also considered. We find the delay behavior may terminate at different parameter areas when the excitation amplitude is fixed and the excitation frequency varies and thus leads to different bursting patterns. Numerical simulations are provided to verify the validity of the study.

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Acknowledgments

The author would like to thank the reviewers and the editor for their helpful comments on the manuscript of this paper.

Funding

The authors would like to thank the National Natural Science Foundation of China (Grant Nos. 61471310 and 61176032) for supporting this research.

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Authors and Affiliations

  1. College of Information Engineering, Xiangtan University, Xiangtan, 411105, Hunan, China

    Zhijun Li, Ying Li, Minglin Ma & Mengjiao Wang

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  1. Zhijun Li
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  2. Ying Li
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Correspondence to Zhijun Li.

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Li, Z., Li, Y., Ma, M. et al. Delayed Transcritical Bifurcation Induced Mixed Bursting in a Modified SM System with Asymmetrically Distributed Equilibria. Braz J Phys 51, 840–849 (2021). https://doi.org/10.1007/s13538-020-00826-y

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