https://doi.org/10.1140/epjp/s13360-024-05118-x
Regular Article
Cascaded bursting oscillations in a Mathieu–van der Pol–Duffing oscillator excited by low-frequency excitations
College of Mechanical Engineering, Zhengzhou University of Science and Technology, 450064, Zhengzhou, China
Received:
17
January
2024
Accepted:
20
March
2024
Published online:
20
April
2024
This paper focuses on bursting behaviors of a Mathieu–van der Pol oscillator excited by low-frequency parametric and external excitations, where two novel bursting modes (namely cascaded bursting oscillations and non-cascaded bursting oscillations) are presented. In particular, the folds induced by the extreme values of the slow variable and the shape of the equilibrium branches are found on the quiescent state vibrations in two bursting patterns. We find that the spiking state vibrations can be induced by the subcritical Hopf bifurcations, supercritical Hopf bifurcations, and limit cycle fold bifurcations, and these three different types of the spiking state attractors can be found simultaneously with the variation of the positive integer ratio n. For the cascaded bursting, more and more large-amplitude limit cycles can be observed with the increase of n, leading to a greater number of the spiking states, while for the non-cascaded bursting, only one large-amplitude limit cycle can be found. In addition, we show that the spiking state attractors cannot be created by the small-amplitude limit cycles due to the Hopf bifurcation delay.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
