Singular perturbations in noisy dynamical systems

Bernard J Matkowsky*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of 'spurious solutions', which led some to conclude that this was 'the failure of MAE'. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

Original languageEnglish (US)
Pages (from-to)570-593
Number of pages24
JournalEuropean Journal of Applied Mathematics
Volume29
Issue number4
DOIs
StatePublished - Aug 1 2018

Fingerprint

Singular Perturbation
Matched Asymptotic Expansions
Dynamical systems
Dynamical system
White noise
Spurious Solutions
Boundary value problems
Mean First Passage Time
Boundary layers
Potential Well
Domain of Attraction
Singularly Perturbed
Boundary Layer
Initial conditions
Collision
Boundary Value Problem

Keywords

  • JWKB methods
  • Singular perturbations
  • asymptotic expansions

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

@article{f0c8abd86ca94e9d9ffd33eed572d9bc,
title = "Singular perturbations in noisy dynamical systems",
abstract = "Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of 'spurious solutions', which led some to conclude that this was 'the failure of MAE'. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.",
keywords = "JWKB methods, Singular perturbations, asymptotic expansions",
author = "Matkowsky, {Bernard J}",
year = "2018",
month = "8",
day = "1",
doi = "10.1017/S0956792518000025",
language = "English (US)",
volume = "29",
pages = "570--593",
journal = "European Journal of Applied Mathematics",
issn = "0956-7925",
publisher = "Cambridge University Press",
number = "4",

}

Singular perturbations in noisy dynamical systems. / Matkowsky, Bernard J.

In: European Journal of Applied Mathematics, Vol. 29, No. 4, 01.08.2018, p. 570-593.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Singular perturbations in noisy dynamical systems

AU - Matkowsky, Bernard J

PY - 2018/8/1

Y1 - 2018/8/1

N2 - Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of 'spurious solutions', which led some to conclude that this was 'the failure of MAE'. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

AB - Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of 'spurious solutions', which led some to conclude that this was 'the failure of MAE'. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

KW - JWKB methods

KW - Singular perturbations

KW - asymptotic expansions

UR - http://www.scopus.com/inward/record.url?scp=85049876751&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049876751&partnerID=8YFLogxK

U2 - 10.1017/S0956792518000025

DO - 10.1017/S0956792518000025

M3 - Article

VL - 29

SP - 570

EP - 593

JO - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

IS - 4

ER -