Long term effects of small random perturbations on dynamical systems

Theoretical and computational tools

Tobias Grafke, Tobias Schäfer, Eric Vanden Eijnden

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. Here we review the theoretical and computational aspects behind these calculations, and propose an algorithm that simplifies the geometric minimum action method to minimize the action in the space of arc-length parametrized curves. We then illustrate this algorithm’s capabilities by applying it to various examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable.

Original languageEnglish (US)
Title of host publicationFields Institute Communications
PublisherSpringer New York LLC
Pages17-55
Number of pages39
Volume79
DOIs
StatePublished - 2017

Publication series

NameFields Institute Communications
Volume79
ISSN (Print)1069-5265

Fingerprint

Random Perturbation
Small Perturbations
Dynamical system
Term
Markov Jump Systems
Stochastic Ordinary Differential Equations
Large Deviation Theory
Markov Jump Processes
Detailed Balance
Reaction Kinetics
Materials Science
Arc length
Multiplicative Noise
Violate
Fluid Dynamics
Computational Methods
Ocean
Atmosphere
Simplify
Partial differential equation

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Grafke, T., Schäfer, T., & Vanden Eijnden, E. (2017). Long term effects of small random perturbations on dynamical systems: Theoretical and computational tools. In Fields Institute Communications (Vol. 79, pp. 17-55). (Fields Institute Communications; Vol. 79). Springer New York LLC. https://doi.org/10.1007/978-1-4939-6969-2_2

Long term effects of small random perturbations on dynamical systems : Theoretical and computational tools. / Grafke, Tobias; Schäfer, Tobias; Vanden Eijnden, Eric.

Fields Institute Communications. Vol. 79 Springer New York LLC, 2017. p. 17-55 (Fields Institute Communications; Vol. 79).

Research output: Chapter in Book/Report/Conference proceedingChapter

Grafke, T, Schäfer, T & Vanden Eijnden, E 2017, Long term effects of small random perturbations on dynamical systems: Theoretical and computational tools. in Fields Institute Communications. vol. 79, Fields Institute Communications, vol. 79, Springer New York LLC, pp. 17-55. https://doi.org/10.1007/978-1-4939-6969-2_2
Grafke T, Schäfer T, Vanden Eijnden E. Long term effects of small random perturbations on dynamical systems: Theoretical and computational tools. In Fields Institute Communications. Vol. 79. Springer New York LLC. 2017. p. 17-55. (Fields Institute Communications). https://doi.org/10.1007/978-1-4939-6969-2_2
Grafke, Tobias ; Schäfer, Tobias ; Vanden Eijnden, Eric. / Long term effects of small random perturbations on dynamical systems : Theoretical and computational tools. Fields Institute Communications. Vol. 79 Springer New York LLC, 2017. pp. 17-55 (Fields Institute Communications).
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