Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations

Xiujing  Han; Yi  Zhang; Qinsheng  Bi; Jürgen  Kurths

doi:10.1063/1.5012519

ABSTRACT

This paper aims to report two novel bursting patterns, the turnover-of-pitchfork-hysteresis-induced bursting and the compound pitchfork-hysteresis bursting, demonstrated for the Duffing system with multiple-frequency parametric excitations. Typically, a hysteresis behavior between the origin and non-zero equilibria of the fast subsystem can be observed due to delayed pitchfork bifurcation. Based on numerical analysis, we show that the stable equilibrium branches, related to the non-zero equilibria resulted from the pitchfork bifurcation, may become the ones with twists and turns. Then, the novel bursting pattern turnover-of-pitchfork-hysteresis-induced bursting is revealed accordingly. In particular, we show that additional pitchfork bifurcation points may appear in the fast subsystem under certain parameter conditions. This creates multiple delay-induced hysteresis behavior and helps us to reveal the other novel bursting pattern, the compound pitchfork-hysteresis bursting. Besides, effects of parameters on the bursting patterns are studied to explore the relation of these two novel bursting patterns.

ACKNOWLEDGMENTS

The authors express their gratitude to the editor and the anonymous reviewers whose comments and suggestions helped in the improvement of this paper. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11572141, 11632008, 11772161, and 11502091) and the Training Project for Young Backbone Teacher of Jiangsu University.

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Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations

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