Slow onset of self-sustained oscillations in a fluctuating sideband-driven electromechanical resonator.

Critical slowing down of the dynamics of a system near bifurcation points leads to long recovery times towards stable states in response to perturbations. Analogously, for systems initially in an unstable state, the relaxation also becomes slow near bifurcation points. Here we explore the onset of self-sustained oscillations in a sideband-driven electromechanical resonator, when the zero-amplitude state changes from stable to unstable. As the system moves away from the unstable zero-amplitude state due to thermal fluctuations, the vibration amplitude increases exponentially with time until nonlinear effects limit the growth and the system settles into stable self-sustained vibrations. We show that the first passage time for the amplitude to reach a threshold value is random and follows a non-Gaussian distribution. On the other hand, the rate of exponential buildup remains constant for different build-up events. As the system approaches a bifurcation point, the build-up of vibrations slows down drastically. The mean and the standard deviation of the first passage time as well as the inverse rate of exponential rise exhibit power law scaling with the distance to either the supercritical or subcritical Hopf bifurcation point with exponent of - 1.