Complex Bursting Patterns in a van der Pol–Mathieu–Duffing Oscillator

Abstract

The pulse-shaped explosion (PSE), characterized by the pulse-shaped quantitative of system solutions varying dramatically, is a special route to bursting oscillations reported recently. This paper reports interesting dynamical behaviors related to the PSE of equilibria, and based on that, the complex bursting dynamics is investigated in a van der Pol–Mathieu–Duffing system with multiple-frequency slow-varying excitations. We find that bifurcations can be observed in a narrow parameter interval within PSE. We also show that two groups of bifurcations are symmetrically arranged on both sides of PSE, and each of which determines a different bursting part. Based on this, two compound bursting patterns, i.e. compound Hopf/Hopf bursting oscillation and compound subHopf/fold cycle bursting oscillation, and a novel type of relaxation oscillation (bursting oscillation of point/point) independent of bifurcations, are revealed. Our results enrich the knowledge of dynamical behaviors related to PSE as well as the possible routes to complex bursting dynamics.

On the occurrence of bursting oscillations in the damping Helmholtz–Rayleigh–Duffing oscillator with slow-changing parametrical and external forcings

“…In general, the large amplitude oscillations are shown as the large-amplitude limit cycle vibrations and the small amplitude oscillations are manifested as the equilibrium vibrations or the small-amplitude limit cycle vibrations [11,12]. An important question is what factors cause the switching between the large-and small-oscillations, bifurcation has been identified as a common switching factor [13,14], some other switching factors have been discovered in recent years, such as bifurcation delay [15,16], canard [17,18] and PSE [19,20].…”

New Burst-Oscillation Mode in Paced One-Dimensional Excitable Systems

“…Since the fast-slow analysis method was proposed by Rinzel in 1985, which can effectively expose the formation and the mechanism of burst oscillation, great achievements have been made in this field. Distinct modes of burst oscillation were identified in different kinds of theoretical models and the corresponding mechanisms were further revealed (de Vries, 1998 ; Perc and Marhl, 2003 ; Zhang et al, 2007 ; Han and Bi, 2011 ; Yang and Hao, 2014 ; Vijay et al, 2019 ; Ma et al, 2021 ; Qian et al, 2021b ). For example, Vries discovered multiple bifurcations of bursting oscillations in a polynomial model (de Vries, 1998 ).…”

“…Vijay et al investigated different transitions of bursting and mixed-mode oscillations in the Liénard system with external sinusoidal forcing (Vijay et al, 2019 ). Ma et al reported the complex bursting dynamics in a van der Pol-Mathieu-Duffing system with multiple-frequency slow-varying excitations and revealed the pulse-shaped explosion as a special route to bursting oscillations (Ma et al, 2021 ). Most of these contributions are devoted to the burst oscillations in the single systems.…”