Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and adaptive material/structure applications such as structural health monitoring and piezoelectric energy harvesting. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors towards the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept towards a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. Using this archetype example, the stochastic normal form of the saddle-node bifurcation is derived from which expressions of the escape statistics are formulated. Non-stationarity is accounted for using a time dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verify that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.
- Aerospace Division
Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation
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Kim, J, Harne, RL, & Wang, KW. "Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation." Proceedings of the ASME 2016 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. Volume 2: Modeling, Simulation and Control; Bio-Inspired Smart Materials and Systems; Energy Harvesting. Stowe, Vermont, USA. September 28–30, 2016. V002T03A003. ASME. https://doi.org/10.1115/SMASIS2016-9051
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