State transitions in the Duffing resonator excited by narrow band random noise

The Dufffing hardening spring resonator has the status of a canonical model for nonlinear vibrations. Over a limited range of excitation frequencies and depending on the degree of nonlinearity, it has three states of response to sinusoidal excitation, one of which is unstable. The system will remain stably in either of the other two states depending on the history of excitation: in a higher energy state for the frequency ascending and in the lower energy state for the frequency descending. When the sinusoidal excitation is replaced by narrow band random excitation, the author showed in a 1961 paper experimentally that the system could make transitions between these two states and argued that the fluctuating phase of the excitation would allow the source to inject or draw energy from the resonator allowing a transition from one state to the other. This presentation develops a dynamic model for the system that allows energy transmission between the source and resonator and an indicator of the state of response based on instantaneous impedance.