Bursting Oscillation and Its Mechanism of a Generalized Duffing–Van der Pol System with Periodic Excitation

The complex bursting oscillation and bifurcation mechanisms in coupling systems of different scales have been a hot spot domestically and overseas. In this paper, we analyze the bursting oscillation of a generalized Duffing–Van der Pol system with periodic excitation. Regarding this periodic excitation as a slow-varying parameter, the system can possess two time scales and the equilibrium curves and bifurcation analysis of the fast subsystem with slow-varying parameters are given. Through numerical simulations, we obtain four kinds of typical bursting oscillations, namely, symmetric fold/fold bursting, symmetric fold/supHopf bursting, symmetric subHopf/fold cycle bursting, and symmetric subHopf/subHopf bursting. It is found that these four kinds of bursting oscillations are symmetric. Combining the transformed phase portrait with bifurcation analysis, we can observe bursting oscillations obviously and further reveal bifurcation mechanisms of these four kinds of bursting oscillations.

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

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.

Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation

The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slow-varying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric “homoclinic/homoclinic” bursting oscillation, symmetric “fold/Hopf” bursting oscillation, symmetric “fold/fold” bursting oscillation, and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop, are revealed with different values of the parameterbby means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency.