The Mechanism of Novel Bursting Patterns in an Externally and Parametrically Excited Hybrid Rayleigh–van der Pol–Duffing System

Due to the existence of the parameter extremum conditions, many practical systems often show some special dynamical behaviors. Analyzing the mechanisms of these special oscillating phenomena is the basis for formulating reasonable control strategies. In this paper, we find that the system solutions approach positive and negative infinities at some critical values in a parametrically and externally excited hybrid Rayleigh–van der Pol–Duffing system. Based on that, we reveal some novel bursting oscillation patterns and investigate their mechanisms by fast–slow analysis method. First, the frequency-domain multiple timescale system can be transformed to an autonomous time-domain multiple timescale system by regarding the low-frequency cosine function as a generalized state variable. Second, the equilibrium points, stabilities and bifurcations are analyzed. We explore the influence of the system parameters on the characteristics of the equilibrium points. In particular, we discuss the phenomenon of the system solutions approaching infinities at the critical values −1 and [Formula: see text]1 and analyze the effects of two key parameters on the dynamical evolutions of the bifurcation structures. Third, the mechanisms of different bursting oscillations are investigated by using time series, phase diagram and superimposed phase diagram. Especially, the influence of the slow passage effect on bursting oscillations is also studied. Finally, the correctness of the results is confirmed by numerical simulations.