Link between externally excited nonlinear system and parametrically excited Duffing oscillator via bursting oscillations and phase transitions

New types of bursting oscillations in the Duffing oscillator with a slow parametric excitation are introduced. By treating low-frequency parametric excitation as a bifurcation parameter in a novel way, two-stage bifurcation is determined. The first is a modified form of a supercritical pitchfork, while the second is supercritical cusp bifurcation. Cusp bifurcation is considered in a topologically equivalent system, where the effect of damping on the creation of three types of bursting oscillations is explained. These three types of oscillations represent a new phenomenon in the Duffing system and here the active and silent phases are clearly distinguished and governable. A new system with external excitation is developed which is equivalent to the parametrically excited Duffing system. In such a system, “bursting like behavior” and its connection with external excitation are established. Also, the base excited real mechanical system and the conditions under which its behavior can be analysed using the developed model are presented. The explanation of the observed phenomena is given based on the first and second phase transitions. Possible applications of the presented approach for low-frequency energy harvesting and detection of external ultra-low frequency excitations and signals are indicated.