Additive noise effects in active nonlinear spatially extended systems
File(s)
Accepted version
Author(s)
Pradas, M
Pavliotis, GA
Kalliadasis, S
Papageorgiou, DT
Tseluiko, D
Type
Journal Article
Abstract
We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multi-scale theory for such systems and apply it to the Kuramoto–Sivashinsky equation as a case study. We first focus on a regime close to the instability onset (primary bifurcation), where the system can be described by a single dominant mode. We show analytically that the resulting noise in the equation describing the amplitude of the dominant mode largely depends on the nature of the stochastic forcing.
For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset so that there are two unstable modes, we observe
numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case.
Key words: Stochastic partial differential equations; Critical phenomena; Multiple scale methods; Singular perturbations
For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset so that there are two unstable modes, we observe
numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case.
Key words: Stochastic partial differential equations; Critical phenomena; Multiple scale methods; Singular perturbations
Date Issued
2012-10-01
Date Acceptance
2012-04-17
Citation
European Journal of Applied Mathematics, 2012, 23 (5), pp.563-591
ISSN
0956-7925
Publisher
Cambridge University Press
Start Page
563
End Page
591
Journal / Book Title
European Journal of Applied Mathematics
Volume
23
Issue
5
Copyright Statement
Copyright © 2012 Cambridge University Press. This article has been published in a revised form in European Journal of Applied Mathematics https://doi.org/10.1017/S0956792512000125. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works.
Subjects
Critical phenomena
DYNAMIC-SYSTEMS
EQUILIBRIUM
EROSION
KURAMOTO-SIVASHINSKY EQUATION
Mathematics
Mathematics, Applied
MODEL
Multiple scale methods
ON-OFF INTERMITTENCY
PARTIAL-DIFFERENTIAL-EQUATIONS
Physical Sciences
QUADRATIC NONLINEARITIES
Science & Technology
Singular perturbations
STABILITY
Stochastic partial differential equations
TRANSITION
Publication Status
Published
Date Publish Online
2012-05-17