https://doi.org/10.1140/epjp/s13360-025-06591-8
Regular Article
Bursting oscillations analysis of the irrational pendulum system
College of Mathematics and Information Science, Hebei University, 071002, Baoding, China
Received:
24
April
2025
Accepted:
23
June
2025
Published online:
10
July
2025
In this paper, we investigate the generation mechanism of bursting oscillations in an irrational pendulum system subjected to a nonlinear periodic damping term and two slow periodic excitations. Regarding the low-frequency external excitation as a slow variable, a fast-slow subsystem is established, and the stability of equilibria and bifurcation condition of the fast-slow subsystem are analyzed. It is found that complex bursting oscillation phenomena occur with the increase of the frequency ratios n of the two slow excitations. Through two-parameter bifurcation diagrams and numerical simulations, we identify distinct bursting patterns: for 
, the asymmetric patterns such as “supHopf/LPC” and “supHopf/supHopf/LPC” emerge; for 
, the symmetric patterns like “supHopf/LPC” and “fold/supHopf/LPC/supHopf/LPC” are observed. The generation of these patterns is related to the changes in bifurcation points and limit cycles. Notably, new fold/Hopf bifurcation points are introduced by increasing the excitation amplitude, leading to more complex interactions of limit cycles. Meanwhile, the influence of fold bifurcations gradually replaces that of limit cycles and Hopf bifurcations on bursting patterns, becoming the dominant factor. The findings of this study offer a theoretical foundation for comprehending the bursting oscillations of irrational pendulum systems.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
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.
