Bursting oscillations, characterized by the alternation between large-amplitude oscillations and small-amplitude oscillations, are complex behaviors of dynamical systems with multiple time scales and have become one of the hot subjects in nonlinear science. Up to now, various underlying mechanisms of bursting oscillations as well as the classifications have been investigated intensively. Recently, a sharp transition behavior, called the "pulsed-shaped explosion (PSE)", was uncovered based on nonlinear oscillators of Rayleigh's type. PSE is characterized by pulse-shaped sharp quantitative changes appearing in the branches of equilibrium point and limit cycle. However, the previous work related to the PSE merely focused on the sharp transitions of unidirectional PSE, and more complex forms of PSE which may lead to more complicated bursting patterns need to be further investigated. Taking a parametrically and externally excited Rayleigh system as an example, we reveal different expression of PSE as well as bursting patterns induced by it. According to the frequency relationship between the two slow excitations, the fast subsystem and the slow variable are obtained by means of frequency-transformation fast-slow analysis. Bifurcation behaviors of the fast subsystem show that, there exist two originally disunited branches of equilibrium point and limit cycle, which extend steeply in different directions and inosculate as a integral structure according to the variation of system parameters. This inosculated integral structure inherits the "steep" properties in different directions from the originally disunited branches, and PSE is thus generated. Unlike the PSE phenomena studied in the previous works, the PSE reported here contains two different peaks in positive and negative directions, which can be named as "positive and negative PSE". Note that the positive and negative PSE essentially complicates bursting dynamics and plays a critical role in bursting. On the other hand, only with the properly chosen parameters could it be created. Based on this, two different types of bursting, i.e., point-point type and cycle-cycle type, are obtained. Subsequently, the transformed phase portraits are introduced to explore dynamical mechanisms of the bursting patterns. We show that, with the variation of the slow parameter, the trajectory may undergo sharp transitions between the rest and active states by positive and negative PSE, and therein lies the generation of bursting. Our results demonstrate the diversity of PSE and give a complement to the underlying mechanisms of bursting.

2019, 51(3): 904-911.
doi: 10.6052/0459-1879-18-319