Multiple pulse-shaped explosions and induced bursting oscillations in the Mathieu-van der Pol-Duffing oscillator

Source abstract

Abstract Bifurcation structure analysis plays an important role in revealing the generation mechanisms of bursting oscillations. In the present work, focusing on the Mathieu-van der Pol-Duffing oscillator subjected to slowly changing harmonic excitations, two bifurcation configurations of a qualitatively different nature: one corresponding to the case where the solution curves of the system are continuous, and the other corresponding to the case where discontinuity points exist on the solution curves, at which the system solutions tend to infinity (exhibiting sharp quantitative changes). We term this phenomenon pulse-shaped explosion (PSE). First, by means of trigonometric transformations and slow-fast dynamical analysis, the system is formulated as a coupling of slow and fast subsystems, where the fast subsystem is further reduced to subsystem (4) and subsystem (5). Then, we carry out solution characteristic analysis and bifurcation analysis for subsystems (4) and (5), respectively, to clarify the underlying mechanism of PSEs and two different types of bifurcation structures, one associated with PSEs and the other independent of PSEs. Finally, based on bifurcation analysis, we present the generation mechanisms of PSE-related and PSE-unrelated bursting oscillations, with particular emphasis on the important roles played by PSEs and the slow passage effect in the production of bursting behaviors. The results of the present work not only provide deeper insight into possible routes to bursting dynamics, but also point out some issues in the nomenclature of bursting oscillation patterns, which may facilitate the exploration of more comprehensive and in-depth classification schemes for bursting oscillations.