Approximation representation of parameterizing manifold and non-Markovian reduced systems for a stochastic Swift–Hohenberg equation

Representation of Manifolds for the Stochastic Swift-Hohenberg Equation with Multiplicative Noise

“…Here, we investigate the corresponding problems for the stochastic Swift-Hohenberg equation with multiplicative noise with pathwise and obtain some new results for it. The results obtained in this paper are different from those in [14], although there are some similar sentences in some manuscripts. It is well known that various noises cause various stochastic processes for stochastic equations with different noises.…”

“…It is well known that there have been some authors to consider the approximation of manifolds in large probability sense [1,2,13] and they have obtained some results until now. In addition, approximation in parameterizing manifold for a stochastic Swift-Hohenberg equation with additive noise have been investigated by us in [14]. Furthermore, it is needed to consider the problems in Stratonovich sense.…”

“…Until now, there have been few consideration from the point of view of approximation in parameterizing manifold under the pathwise sense for the stochastic Swift-Hohenberg equation with multiplicative noise. The ideas in [14] can be used to consider the approximation of manifold for some stochas-tic equations with multiplicative noise. Because the different difficulties come from different noise terms, there are some different methods and techniques in studying stochastic equations with multiplicative noise.…”

“…It is well known that various noises cause various stochastic processes for stochastic equations with different noises. The main differences from results in [14] are given by some formulas with various mathematics meanings, in which some different stochastic functions are used. Because the different difficulties mainly come from various noise terms, there are some new difficulties coming from the multiplicative noise solved in our manuscript.…”

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

“…By using the expression of û (2) s given in (30), we are now in position to derive an operational version of (14) for the modeling of SPDE dynamics projected onto the resolved mode. In that aspect, let us denote (t, ) ∶= 1 (t, )e 1 + 2 (t, )e 2 with i (t, ) ∶=< (t, ),…”