Optimal paths, caustics, and boundary layer approximations in stochastically perturbed dynamical systems

Robert S. Maier, Daniel L. Stein

    Research output: Chapter in Book/Report/Conference proceedingChapter

    Abstract

    We study the asymptotic properties of overdamped dynamical systems with one or more point attractors, when they are perturbed by weak noise. In the weak-noise limit, fluctuations to the vicinity of any specified non-attractor point will increasingly tend to follow a well-defined optimal path. We compute precise asymptotics for the frequency of such fluctuations, by integrating a matrix Riccati equation along the optimal path. We also consider noise-induced transitions between domains of attraction, in two-dimensional double well systems. The optimal paths in such systems may focus, creating a caustic. We examine `critical' systems in which a caustic is beginning to form, and show that due to criticality, the mean escape time from one well to the other grows in the weak-noise limit in a non-exponential way. The analysis relies on a Maslov-WKB approximation to the solution of the Smoluchowski equation.

    Original languageEnglish (US)
    Title of host publication15th Biennial Conference on Mechanical Vibration and Noise
    Pages903-910
    Number of pages8
    Edition3 Pt A/2
    StatePublished - 1995
    EventProceedings of the 1995 ASME Design Engineering Technical Conference - Boston, MA, USA
    Duration: Sep 17 1995Sep 20 1995

    Publication series

    NameAmerican Society of Mechanical Engineers, Design Engineering Division (Publication) DE
    Number3 Pt A/2
    Volume84

    Other

    OtherProceedings of the 1995 ASME Design Engineering Technical Conference
    CityBoston, MA, USA
    Period9/17/959/20/95

    ASJC Scopus subject areas

    • Engineering(all)

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  • Cite this

    Maier, R. S., & Stein, D. L. (1995). Optimal paths, caustics, and boundary layer approximations in stochastically perturbed dynamical systems. In 15th Biennial Conference on Mechanical Vibration and Noise (3 Pt A/2 ed., pp. 903-910). (American Society of Mechanical Engineers, Design Engineering Division (Publication) DE; Vol. 84, No. 3 Pt A/2).