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

Jinki Kim, R. L. Harne, K. W. Wang

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


Accurately predicting the onset of large behavioral deviations associated with saddlenode 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 everpresent 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.

Original languageEnglish (US)
Article number011009
JournalJournal of Computational and Nonlinear Dynamics
Issue number1
StatePublished - Jan 1 2017

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Mechanical Engineering
  • Applied Mathematics


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