Eur. Phys. J. Spec. Top. (2022) 231: 2249-2263
https://doi.org/10.1140/epjs/s11734-022-00485-x

Regular Article

Novel bursting patterns induced by hysteresis loops in a one-degree-of-freedom nonlinear oscillator with parametric and external excitations

Faculty of Civil Engineering and Mechanics, Jiangsu University, 212013, Zhenjiang, People’s Republic of China

^a 1000004942@ujs.edu.cn

Received: 23 November 2021
Accepted: 17 February 2022
Published online: 5 March 2022

Abstract

This paper investigates the novel bursting patterns induced by different types of hysteresis loops in a parametrically and externally driven one-degree-of-freedom nonlinear oscillator (abbreviated as the PEDODOFNO) that can be represented the dynamics of a density perturbation in plasma device model. The bursting behavior named “fold/turning point-turning point/fold” form via “fold/turning point” hysteresis loop is investigated using the slow–fast analysis method in detail. Moreover, the dynamical transitions of the novel bursting, i.e., bursting of “fold/turning point-turning point/extreme point” form via “fold/turning point” hysteresis loop, bursting of “fold/fold-fold/extreme point” form via “fold/fold” hysteresis loop, bursting of “fold/fold-fold/extreme point-extreme point/turning point-turning point/fold” form via “fold/fold-extreme point/turning point” hysteresis loop, bursting of “fold/fold-fold/extreme point-extreme point/fold-fold/fold” form via “fold/fold-extreme point/fold” hysteresis loop, bursting of “fold/fold-fold/turning point-turning point/fold” form via “fold/fold” hysteresis loop, and bursting of “fold/fold” form via “fold/fold” hysteresis loop with a hypocritical cycle, are also studied. In these bursting patterns, the excited states are not the oscillations of the trajectory vibrating in the stable limit cycles conventionally, but the oscillations behaving in the hypocritical cycles that are produced by the procedures of the trajectory running steadily to the control range of the stable equilibria. The hypocritical cycles disappear by two approaches: the first is colliding with the saddle branches and the second is the control parameter reaching the maxima value. The two approaches result in the occurrence of the turning point and the extreme point, and then, the different types of hysteresis loops related to fold bifurcation, the turning point and the extreme point are created, which lead to the appearance of the distinctive hysteresis loops.