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Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol’s equation

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Abstract

This paper investigates the emergence of mixed-mode oscillations (MMOs) in the forced Van der Pol’s equation. It is found that the MMOs studied here can be classified as a slow passage through canard explosion, which is different from the usual fast-slow bursters. We first consider the external forcing as a control parameter and study its influence on the Van der Pol’s equation with constant forcing (VPCF). Then we briefly discuss the famous canard phenomenon in VPCF. The results of these analysis, together with the “transformed phase diagram,” are applied to the forced Van der Pol’s equation, which shows that the canard explosion and the external forcing plays an important role in the generation of MMOs, that is, the MMOs are created since the external forcing slowly and periodically visits the rest and spiking areas of VPCF.

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

    Xiujing Han & Qinsheng Bi

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  1. Xiujing Han
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  2. Qinsheng Bi
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Correspondence to Qinsheng Bi.

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Han, X., Bi, Q. Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol’s equation. Nonlinear Dyn 68, 275–283 (2012). https://doi.org/10.1007/s11071-011-0226-9

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  • DOI: https://doi.org/10.1007/s11071-011-0226-9

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