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Predicting non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems

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Abstract

Saddle-node bifurcation can cause dynamical systems undergo large and sudden transitions in their response, which is very sensitive to stochastic and non-stationary influences that are unavoidable in practical applications. Therefore, it is essential to simultaneously consider these two factors for estimating critical system parameters that may trigger the sudden transition. Although many systems exhibit non-smooth dynamical behavior, estimating the onset of saddle-node bifurcation in them under the dual influence remains a challenge. In this work, a new theoretical framework is developed to provide an effective means for accurately predicting the probable time at which a non-smooth system undergoes saddle-node bifurcation while the governing parameters are swept in the presence of noise. The stochastic normal form of non-smooth saddle-node bifurcation is scaled to assess the influence of noise and non-stationary factors by employing a single parameter. The Fokker–Planck equation associated with the scaled normal form is then utilized to predict the distribution of the onset of bifurcations. Experimental efforts conducted using a double-well Duffing analog circuit successfully demonstrate that the theoretical framework developed in this study provides accurate prediction of the critical parameters that induce non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems.

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Acknowledgements

This research was partially supported by the National Science Foundation under Award Nos. 1232436 and 1661568 and the University of Michigan Collegiate Professorship fund.

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  1. Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

    Jinki Kim & K. W. Wang

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  1. Jinki Kim
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Correspondence to Jinki Kim.

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Kim, J., Wang, K.W. Predicting non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems. Nonlinear Dyn 93, 251–258 (2018). https://doi.org/10.1007/s11071-018-4189-y

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