Abstract
Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.
Highlights
The Swift-Hohenberg equation was originally derived by Swift and Hohenberg[27] as a model for convective instability in Rayleigh-Benard convection
It is difficult to stabilize the control parameters to the precision of the noise strength, which is extremely small in the case of thermal fluctuations
The effects of thermal fluctuations on the onset of convective motion needs to be taken into account, and
Summary
The Swift-Hohenberg equation was originally derived by Swift and Hohenberg[27] as a model for convective instability in Rayleigh-Benard convection. Swift and Hohenberg [27]’s original model is a stochastic partial differential equation(SPDE). The purpose of the present paper is to study random invariant manifolds for the stochastic Swift-Hohenberg equation (1), together with their approximations and geometric shapes. Stochastic partial differential equations (SPDEs) play important roles in modeling, analyzing, simulating and predicting complex phenomena under random fluctuation in various fields [19, 13, 31]. We need a random transformation to convert an SPDE to a partial differential equation random coefficients To this end, we consider the following scalar Langevin equation dz + zdt = σdW.
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