### Abstract

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point S. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength ∈, the system state will eventually leave the domain of attraction Ωof S. We analyze the case when, as ∈ → 0, the exit location on the boundary ∂Ωis increasingly concentrated near a saddle point H of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on ∂Ω is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter μ equal to the ratio |λ_{s}(H)|/λ_{u}(H) of the stable and unstable eigenvalues of the linearized deterministic flow at H. If μ< 1, then the exit location distribution is generically asymptotic as ∈ → 0 to a Weibull distribution with shape parameter 2/μ, on the script O sign(∈^{μ/2}) lengthscale near H. If μ > 1, it is generically asymptotic to a distribution on the script O sign(∈^{1/2}) lengthscale, whose moments we compute. Our treatment employs both matched asymptotic expansions and stochastic analysis. As a byproduct of our treatment, we clarify the limitations of the traditional Eyring formula for the weak-noise exit time asymptotics.

Original language | English (US) |
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Pages (from-to) | 752-790 |

Number of pages | 39 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 57 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1997 |

Externally published | Yes |

### Keywords

- Ackerberg - O'malley resonance
- Exit location
- First passage time
- Large deviations
- Large fluctuations
- Matched asymptotic expansions
- Saddle point avoidance
- Singular perturbation theory
- Stochastic analysis
- Stochastic exit problem
- Wentzell-Freidlin theory

### ASJC Scopus subject areas

- Applied Mathematics

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## Cite this

*SIAM Journal on Applied Mathematics*,

*57*(3), 752-790. https://doi.org/10.1137/S0036139994271753