Stability of two-mode internal resonance in a nonlinear oscillator

We analyze the stability of synchronized periodic motion for two coupled oscillators, representing two interacting oscillation modes in a nonlinear vibrating beam. The main oscillation mode is governed by the forced Duffing equation, while the other mode is linear. By means of the multiple-scale approach, the system is studied in two situations: an open-loop configuration, where the excitation is an external force, and a closed-loop configuration, where the system is fed back with an excitation obtained from the oscillation itself. The latter is relevant to the functioning of time-keeping micromechanical devices. While the accessible amplitudes and frequencies of stationary oscillations are identical in the two situations, their stability properties are substantially different. Emphasis is put on resonant oscillations, where energy transfer between the two coupled modes is maximized and, consequently, the strong interdependence between frequency and amplitude caused by nonlinearity is largely suppressed.