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On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

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Abstract

The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol–Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slow-varying parameter. The equilibrium branches may undergo different types of bifurcations, such as Hopf and pitchfork bifurcations. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit, may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously different amplitudes. At the pitchfork bifurcation point, two possible jumping ways may result in two coexisted asymmetric bursting attractors, which may expand in the phase space to interact with each other to form an enlarged symmetric bursting attractor with doubled period. The inertia of the movement along the stable equilibrium may cause the trajectory to pass across the related bifurcations, leading to the delay effect of the bifurcations. Not only the large exciting amplitude, but also the large value of the exciting frequency may increase inertia of the movement, since in both the two cases, the change rate of the slow-varying parameter may increase. Therefore, a relative small exciting frequency may be taken in order to show the possible influence of all the equilibrium branches and their bifurcations on the dynamics of the full system.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The work is supported by the National Natural Science Foundation of China (Grant Nos. 11872188,11632008).

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Authors and Affiliations

  1. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, 212013, China

    Xiaofang Zhang, Bin Zhang, Xiujing Han & Qinsheng Bi

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  1. Xiaofang Zhang
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Correspondence to Qinsheng Bi.

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No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.

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Zhang, X., Zhang, B., Han, X. et al. On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation. Nonlinear Dyn 108, 2097–2114 (2022). https://doi.org/10.1007/s11071-022-07309-6

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