On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Abstract

The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol-Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slowvarying parameter, on which different types of bifurcations, such as Hopf and pitch fork bifurcations, can be observed. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously dif-

Bursting dynamics and the bifurcation mechanism of a modified Rayleigh-van der Pol-Duffing oscillator

“…When the trajectory reaches at the supercritical Hopf point, because of the delay behavior, QS1 lasts until δ = 1.0. It is worth noting that by changing the excitation frequency, the delay time will also change accordingly, which has been confirmed in the previous study [39]. Figure 5(d) reveals the return trajectory.…”