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Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools

  • Tobias GrafkeEmail author
  • Tobias Schäfer
  • Eric Vanden-Eijnden
Chapter
Part of the Fields Institute Communications book series (FIC, volume 79)

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.

Notes

Acknowledgements

We would like to thank M. Cates, A. Donev, and F. Bouchet for helpful discussions. This research was supported in part by the NSF grants DMR-1207432 (T. Grafke), DMS-1108780 and DMS-1522737 (T. Schäfer), and DMS-1522767 (E. Vanden-Eijnden).

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Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Tobias Grafke
    • 1
    Email author
  • Tobias Schäfer
    • 2
    • 3
  • Eric Vanden-Eijnden
    • 1
  1. 1.Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsCollege of Staten Island 1S-215Staten IslandUSA
  3. 3.Physics Program at the CUNY Graduate CenterNew YorkUSA

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