Normal form transforms separate slow and fast modes in stochastic dynamical systems

Article


Roberts, A. J.. 2008. "Normal form transforms separate slow and fast modes in stochastic dynamical systems." Physica A: Statistical Mechanics and its Applications. 387 (1), pp. 12-38. https://doi.org/10.1016/j.physa.2007.08.023
Article Title

Normal form transforms separate slow and fast modes in stochastic dynamical systems

ERA Journal ID351
Article CategoryArticle
Authors
AuthorRoberts, A. J.
Journal TitlePhysica A: Statistical Mechanics and its Applications
Journal Citation387 (1), pp. 12-38
Number of Pages27
Year2008
Place of PublicationNetherlands
ISSN0378-4371
1873-2119
Digital Object Identifier (DOI)https://doi.org/10.1016/j.physa.2007.08.023
Web Address (URL)http://www.sciencedirect.com/science/article/pii/S0378437107008655
Abstract

Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics. We explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to separate \emph{all} slow processes from \emph{all} fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. Some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system. The results will help us accurately model, interpret and simulate multiscale stochastic systems.

Keywordsstochastic dynamical systems; multiscale modelling
ANZSRC Field of Research 2020490510. Stochastic analysis and modelling
499999. Other mathematical sciences not elsewhere classified
490409. Ordinary differential equations, difference equations and dynamical systems
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Byline AffiliationsComputational Engineering and Science Research Centre
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