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Fast estimation of bifurcation conditions using noisy response data

In this work we consider the effects of noise on system behavior during parameter sweeps through bifurcations that induce sharp jumps in response amplitudes. These problems arise in a variety of applications, including the use of bifurcation amplification in micro-sensors. Due to inherent system noise, the observed bifurcation events are stochastic in nature and one must estimate parameter values from a distribution. A stochastic dynamical systems analysis allows one to distill the problem to a one-dimensional, one-parameter Fokker-Planck equation, where the parameter is the ratio of the noise intensity to sweep rate. Approximate closed form solutions for the distributions are obtained in the limits of slow and fast parameter sweep rates, and a numeric solution captures the intermediate sweep range that bridges these two approximations. These results are essential for quantifying errors in bifurcation amplifiers and for optimizing bifurcation detection schemes, as used in sensing applications. Preliminary experimental results for a parametrically excited microdevice show good qualitative agreement with the theory.