Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation

Abstract

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 microscale mass sensing. 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 toward 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 toward 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. The stochastic normal form of the saddle-node bifurcation is derived from the governing equation of this oscillator to formulate the probability distribution of escape events. 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 (PDF) to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verifies that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

Online Signal Denoising Using Adaptive Stochastic Resonance in Parallel Array and Its Application to Acoustic Emission Signals

“…The bistable circuit involves saddle-node bifurcation that activates sudden transitions between the intrawell and interwell oscillations. As a result, bistable circuits have been applied for various bifurcation-based applications [37,39,[55][56][57][58][59]. Since this bistable circuit design exhibits negligible backward coupling due to the characteristics of op-amp, it is suitable to utilize as a signal conditioning device that has little effect on the input signal [37].…”

“…Since the optimal noise intensity (D opt = 0.72) determined in Sec. 2 is the optimal value for a non-dimensional bistable Duffing system, we approximated the bistable circuit as a double-well Duffing oscillator [56] and obtained the optimal noise intensity ( D opt = 0.5 V 2 ) for the bistable circuit by a series of back calculations. A more detailed discussion is given in Appendix B.…”

Electromechanical impedance-based damage identification enhancement using bistable and adaptive piezoelectric circuitry

“…Overview of impedance change measurement using a network of bistable circuits Figure 2(a) presents the configuration of a beam structure integrated with the proposed bistable and adaptive piezoelectric circuitry. The double-well Duffing analog circuit 34,36,37 employed as bistable circuit in this study (Figure 2(b)) involves saddle-node bifurcation that is activated when the input voltage amplitude exceeds a critical threshold, while the excitation frequency is fixed at a value that is below the linear resonance frequency of the circuit. As a result, the circuit undergoes sudden and dramatic change in the output voltage level with minute input voltage variation around the critical threshold.…”

“…The stochastic and non-stationary influences on the activation of classical smooth saddle-node bifurcations can be investigated by utilizing the stochastic normal form of dynamic saddle-node bifurcation, which is a simplified form that describes the system dynamics local to the bifurcating point. 36,43 Since the bistable circuit exhibits non-smooth dynamical behavior due to the non-linear feedback loop among an op-amp and a pair of diodes, we develop a normal form of non-smooth saddle-node bifurcation to examine the local dynamics near the bifurcation point 44…”

“…The horizontal axis is the input voltage amplitude and the cumulative probability is provided along the vertical axis. If the gain value is designed such that the input voltage amplitude V i is swept up to V A in Figure 4, which is approximately two standard deviations SD V esc below the mean value V esc , 36 the bistable circuit will undergo saddle-node bifurcation from intra- to interwell oscillations with 2.5% probability. In other words, the bistable circuit will have 97.5% probability to exhibit intrawell oscillation.…”