The formation of the pulse-shaped-explosion type bursting dynamics in a van der pol-duffing-rayleigh system

In many nonlinear systems, the variations of the system parameters may cause sharp transitions in the system solution curves, which have been proven to be an important route to bursting dynamics. This paper presents three different forms of the sharp transitions caused by the changes of the system parameters, known as the pulse-shaped-explosion (PSE), and investigates the generation of the bursting behaviors triggered by the combined action of the PSE and bifurcation structures. The occurrence of the PSE phenomenon is closely related to the system parameter values, which is characterized by the solution curve of the dynamical equation inclining to the infinity at some discontinuity points. We find that the sharp transitions related to the changes of the system parameters can be observed in the solution curves, and these sharp transitions tend towards the positive or negative infinities, which give rise to the formation of the PSE. Meanwhile, we note that the system parameters unrelated to the system solutions may be related to the bifurcation types and structures, and the variations of these parameters can lead to the different bifurcations structures. The coupling action of the PSE and different bifurcation structures may trigger the system to demonstrate a variety of alternating forms of large-amplitude and small-amplitude oscillations, thereby inducing different bursting dynamics. The results are of great significance for explaining the formation of the special bursting dynamics and expanding the applications of the slow fast dynamics method. At the same time, the results also provide necessary theoretical support for the system parameter selection, identification, and control strategy formulation.