### 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 language | English (US) |
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Title of host publication | 15th Biennial Conference on Mechanical Vibration and Noise |

Pages | 903-910 |

Number of pages | 8 |

Volume | 84 |

Edition | 3 Pt A/2 |

State | Published - 1995 |

Event | Proceedings of the 1995 ASME Design Engineering Technical Conference - Boston, MA, USA Duration: Sep 17 1995 → Sep 20 1995 |

### Other

Other | Proceedings of the 1995 ASME Design Engineering Technical Conference |
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City | Boston, MA, USA |

Period | 9/17/95 → 9/20/95 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

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 -