TY - CHAP

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.

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M3 - Chapter

AN - SCOPUS:0029431104

T3 - American Society of Mechanical Engineers, Design Engineering Division (Publication) DE

SP - 903

EP - 910

BT - 15th Biennial Conference on Mechanical Vibration and Noise

T2 - Proceedings of the 1995 ASME Design Engineering Technical Conference

Y2 - 17 September 1995 through 20 September 1995

ER -