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Research Papers

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

[+] Author and Article Information
Jinki Kim

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: jinkikim@umich.edu

R. L. Harne

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210

K. W. Wang

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 3, 2016; final manuscript received June 28, 2016; published online September 1, 2016. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 12(1), 011009 (Sep 01, 2016) (9 pages) Paper No: CND-16-1053; doi: 10.1115/1.4034128 History: Received February 03, 2016; Revised June 28, 2016

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.

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Figures

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Fig. 3

(a) Analog circuit diagram employed in the theoretical and experimental investigation. (b) Experimentally measured (squares) nonlinear voltage function with respect to output voltage amplitude. The cubic polynomial fit (solid curve) has R2 fitness of 0.866.

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Fig. 2

Experimentally measured output voltage amplitudes of the circuit (a) as input voltage amplitude sweep rate varies and (b) in the presence of different levels of additive Gaussian white noise with a fixed sweep rate of 0.1 V/s. The critical input voltage amplitudes that activate bifurcations are presented for each case.

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Fig. 1

Potential energy of a double-well Duffing oscillator (solid curve). Illustrative dynamic trajectories for (a) intrawell and (b) interwell steady-state oscillations.

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Fig. 4

Probability density of escape time T when α = 0.1

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Fig. 5

The polynomial fit ke(T − μT)2 (solid) of F(T) (dotted curve)

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Fig. 6

The analytical approximation (solid curve) of the mean escape time is presented with the results obtained by numerically solving the PDF (dashed curve), Monte-Carlo simulation (square), and experimental measurement (circle) with respect to the scaled noise level α

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Fig. 7

Comparison of analytical, numerical, and experimental results of activating the dynamic saddle-node bifurcation of the double-well Duffing circuit. For (a) the input voltage amplitude is swept at rate of 0.1 V/s with 0.015 mV additive noise level, while for (b) the sweep rate is 2.5 V/s using 10 mV level noise. Analytically estimated mean escape input voltage amplitude 〈Vesc〉 for both cases is presented as dotted vertical lines.

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Fig. 8

Experimentally measured (data points with solid curves) and analytically estimated (dotted curves) mean escape input voltage amplitudes are presented with respect to the inverse of the scaled noise level α for each different additive noise level given next to each curve

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