On the low-dimensional modelling of Stratonovich stochastic differential equations

Xu, Chao; Roberts, A. J.

doi:10.1016/0378-4371(95)00387-8

Cited by 38 publications

(44 citation statements)

References 25 publications

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“…Next we give an asymptotic expansion of h 0 (x, ω) by (26). First we need a result on the approximation in the sense of distribution to some stochastic convolution as ϵ → 0 [24].…”

Section: Application To Example Systemsmentioning

confidence: 99%

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Slow manifold and averaging for slow–fast stochastic differential system

Wang

Roberts

2013

Journal of Mathematical Analysis and Applications

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a b s t r a c tWe consider stochastic dynamical systems with multiple time scales. An intermediate reduced model is obtained and explored for a slow-fast system when the fast mode is driven by white noise. First, and for later comparison, one approximation to the reduced stochastic system on the exponentially attracting stochastic slow manifold is derived to errors of order O(ϵ). Second, because the noise only drives the fast modes, averaging derives an autonomous deterministic system with errors of order O( √ ϵ). Then a martingale argument accounts for fluctuations about the averaged system to form an intermediate reduced model with errors of order O(ϵ)-the autonomous deterministic system now driven by white noise. This intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold.These results not only connect averaging with the stochastic slow manifold, they also provide a martingale method for improving averaged models of stochastic systems.

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“…Next we give an asymptotic expansion of h 0 (x, ω) by (26). First we need a result on the approximation in the sense of distribution to some stochastic convolution as ϵ → 0 [24].…”

Section: Application To Example Systemsmentioning

confidence: 99%

“…For any Lipschtiz continuous function h : R n → R m and for any X ∈ R n , t ≥ 0, consider the solution (X ϵ (t, ω, h(X )),Ȳ ϵ (t, ω, h(X ))) to (23)- (24) with initial value (X, h(X )). By Lemma 2.2, a direct computation yields…”

Section: Definition 33mentioning

confidence: 99%

Slow manifold and averaging for slow–fast stochastic differential system

Wang

Roberts

2013

Journal of Mathematical Analysis and Applications

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“…for some effectively new noiseφ(t) (Chao & Roberts 1996). Such replacement was also justified by Khasminskii (1996) as described by Sri Namachchivaya & Leng (1990).…”

Section: ) (Principle 3 and Principle 4)mentioning

confidence: 92%

“…The quadratic noise term in (65) generates a mean drift and an effective new noise over long times: Roberts (2006c) and Chao & Roberts (1996) argued that over long times…”

Section: Compare With Averagingmentioning

confidence: 99%

“…The aim underlying all the exploration in this article is to derive normal forms useful for macroscopic modelling of stochastic systems when the systems are specified at a detailed microscopic level. Thus we endeavour to separate all fast time processes from the slow processes (Chao & Roberts 1996, Roberts 2006c. Such separation is especially interesting in stochastic systems as white noise has fluctuations on all time scales.…”

Section: Introductionmentioning

confidence: 99%

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

Roberts

2008

Physica A: Statistical Mechanics and its Applications

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122

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Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from 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 all slow processes from all fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. The results will help us accurately model, interpret and simulate multiscale stochastic systems.

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Stochastic attractor bifurcation for the two‐dimensional Swift‐Hohenberg equation

Hernández

Ong

2018

Math Methods in App Sciences

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The main objectives of this article are to introduce stochastic parameterizing manifolds and to study the dynamical transitions of the two‐dimensional stochastic Swift‐Hohenberg equation. The study is based on the general framework developed by Chekroun, Liu and Wang. The detailed effect of the noise on the transition and on the stochastic low‐dimensional parameterization of the system is obtained.

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On the low-dimensional modelling of Stratonovich stochastic differential equations

Cited by 38 publications

References 25 publications

Slow manifold and averaging for slow–fast stochastic differential system

Slow manifold and averaging for slow–fast stochastic differential system

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

Stochastic attractor bifurcation for the two‐dimensional Swift‐Hohenberg equation

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