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.