Bursting Oscillations Induced by Coexisted Cycles Separated by Fold Limit Cycle Bifurcation

This study identifies the basic characteristics as well as the bifurcation mechanism of slow-fast behaviors in a typical model with two components for vibration utilization or mitigation.

A modified slow-fast analysis method is employed, in which the whole slow-varying excitation is regarded as a parameter. Based on the bifurcation analysis of the generalized autonomous fast subsystem, the mechanism for different bursting oscillators can be obtained by overlapping the so-called transformed phase portrait and the behaviors of the fast subsystem.

Coexisted attractors in the fast subsystem may influence the dynamics of the full system, leading to different bursting oscillations. Stable cycle relate to the quasi-periodic movement, is separated from the stable cycle associated with the bursting by an unstable cycle, yielding a threshold between the trajectories of the two attractors. Delay effect of bifurcation may cause the disappearance of some stable attractors of the fast subsystem with short windows on the bursting attractor.

For the coupled oscillator in real engineering applications, bursting oscillations may take place when coupling of two scales is introduced. Coexisted attractors influence the trajectory of the full system in turn, resulting in a few interesting behaviors of the oscillations.