Nonlinear Dynamics

, Volume 77, Issue 4, pp 1285–1296 | Cite as

Study of mixed-mode oscillations in a parametrically excited van der Pol system

Original Paper
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Abstract

This paper investigates the effects of slowly varying parametric excitation on the dynamics of van der Pol system. Periodic bifurcation delay behaviors are exhibited when the parametric excitation slowly passes through Hopf bifurcation value of the controlled van der Pol system. The first bifurcation delay behavior relies on initial conditions, while the bifurcation delay behaviors that follow the first one are immune to initial conditions. These bifurcation delay behaviors result in a hysteresis loop between the spiking attractor and the rest state, which is responsible for the generation of mixed-mode oscillations. Then an approximate calculation for the number of spikes in each cluster of repetitive spiking of mixed-mode oscillations is explored based on bifurcation delay behaviors. Theoretical results agree well with numerical simulations.

Keywords

Slowly varying parametric excitation Mixed-mode oscillations Bifurcation delay  Number of spikes 
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Notes

Acknowledgments

The authors express their gratitude to the reviewers whose comments help the improvements of this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11202085, 21276115 and 11302087), the Natural Science Foundation of Jiangsu Province (Grant No. BK20130479), the Research Foundation for Advanced Talents of Jiangsu University (Grant No. 11JDG075), the Scientific Research Innovation Foundation of Jiangsu Province (Grant No. CXZZ_0653) and the NSF of Higher Education Institutions of Jiangsu Province (No. 13KJB110005).

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Faculty of Civil Engineering and MechanicsJiangsu UniversityZhenjiangPeople’s Republic of China

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