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 proceedingConference contribution

    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
    Volume84
    Edition3 Pt A/2
    StatePublished - 1995
    EventProceedings of the 1995 ASME Design Engineering Technical Conference - Boston, MA, USA
    Duration: Sep 17 1995Sep 20 1995

    Other

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

    Fingerprint

    Riccati equations
    Dynamical systems
    Boundary layers

    ASJC Scopus subject areas

    • Engineering(all)

    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., Vol. 84, pp. 903-910)

    Optimal paths, caustics, and boundary layer approximations in stochastically perturbed dynamical systems. / Maier, Robert S.; Stein, Daniel L.

    15th Biennial Conference on Mechanical Vibration and Noise. Vol. 84 3 Pt A/2. ed. 1995. p. 903-910.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Maier, RS & Stein, DL 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 edn, vol. 84, pp. 903-910, Proceedings of the 1995 ASME Design Engineering Technical Conference, Boston, MA, USA, 9/17/95.
    Maier RS, Stein DL. 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. Vol. 84. 1995. p. 903-910
    Maier, Robert S. ; Stein, Daniel L. / Optimal paths, caustics, and boundary layer approximations in stochastically perturbed dynamical systems. 15th Biennial Conference on Mechanical Vibration and Noise. Vol. 84 3 Pt A/2. ed. 1995. pp. 903-910
    @inproceedings{2fb8cc07f9c54a9893b4789d9a97404a,
    title = "Optimal paths, caustics, and boundary layer approximations in stochastically perturbed dynamical systems",
    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.",
    author = "Maier, {Robert S.} and Stein, {Daniel L.}",
    year = "1995",
    language = "English (US)",
    volume = "84",
    pages = "903--910",
    booktitle = "15th Biennial Conference on Mechanical Vibration and Noise",
    edition = "3 Pt A/2",

    }

    TY - GEN

    T1 - Optimal paths, caustics, and boundary layer approximations in stochastically perturbed dynamical systems

    AU - Maier, Robert S.

    AU - Stein, Daniel L.

    PY - 1995

    Y1 - 1995

    N2 - 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.

    AB - 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.

    UR - http://www.scopus.com/inward/record.url?scp=0029431104&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0029431104&partnerID=8YFLogxK

    M3 - Conference contribution

    VL - 84

    SP - 903

    EP - 910

    BT - 15th Biennial Conference on Mechanical Vibration and Noise

    ER -