Bursting patterns in a tri-stable oscillator under parametric excitations

The motion of the tri-stable oscillator can be transferred between different equilibrium points to produce a large response, which is the desire of vibration energy harvesting technology. Therefore, the vibration of the tri-stable system has been a widespread concern. However, the mechanism of the system with parametric slow excitations has not been discovered. To study this issue, we treat a novel bursting pattern in a tri-stable oscillator under slow cosinoidal excitations. The complex dynamic behaviors of bursting oscillation under single and two frequencies are presented using the fast-slow analysis method, the one-parameter bifurcation diagrams are plotted, and the bifurcation patterns associated with independent loop and closed loop are observed. Furthermore, different types of bursting oscillations induced by subcritical pitchfork/fold bifurcations are presented, and jumping phenomena and hysteresis behaviors are displayed. Moreover, the stabilities of equilibrium of the independent and closed loop positions are checked by potential energy, and the transition of different equilibrium configurations is validated by introducing the transformed phase diagram. Finally, the attraction basin of each equilibrium point is estimated via cell mapping, and the multivalued response orbits are recognized.