Compound Bursting Behaviors in the Parametrically Amplified Mathieu–Duffing Nonlinear System

Abstract

This paper primarily investigates five novel compound bursting patterns in the parametrically amplified Mathieu-Duffing nonlinear energy harvesters (PAMDNEH). The five new compound bursting patterns, i.e. compound “subHopf/LPC-delayed subHopf/LPC-subHopf/LPC” bursting, compound “subHopf/LPC-subHopf/LPC” bursting, compound “subHopf/LPC-subHopf/LPC” bursting via “subHopf/subHopf” hysteresis loop, compound “subHopf/supHopf-supHopf/LPC” bursting via “subHopf/subHopf” hysteresis loop and “subHopf/LPC” bursting via “subHopf/subHopf” hysteresis loop, are explored. In addition, two normal bursting patterns, i.e. “subHopf/subHopf” bursting and “fold/fold” bursting, are also investigated. By using the two-bifurcation analysis method, the compound “subHopf/LPC-delayed subHopf/LPC-subHopf/LPC” bursting and its dynamical evolutions is investigated. The correctness of the results is verified by the numerical simulation. We find that the parameter plane can be divided into six different areas according to the types and numbers of the Hopf bifurcation points, and different bifurcation structures and different patterns of the clusters will be created, which lead to different bursting oscillations, including compound bursting types and normal bursting patterns. Because bursting oscillations is created when the variables switch between/among different attractors, the bursting oscillation amplitude is big enough to surmount the potential obstacles of the energy harvesters. Therefore, bursting oscillations may have a very important application in energy harvesting.