Wong–Zakai approximations and limiting dynamics of stochastic Ginzburg–Landau equations

2021 ◽  
Author(s):  
Ji Shu ◽  
Dandan Ma ◽  
Xin Huang ◽  
Jian Zhang
Keyword(s):  
White Noise ◽  
Landau Equation ◽  
Upper Semicontinuity ◽  
Approximate Equation ◽  
Ginzburg Landau ◽  
Random System ◽  
Additive White Noise ◽  
Random Attractors ◽  
Ginzburg Landau Equations ◽  
Weaker Conditions

This paper deals with the Wong–Zakai approximations and random attractors for stochastic Ginzburg–Landau equations with a white noise. We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation. In addition, when the stochastic Ginzburg–Landau equation is driven by an additive white noise, we establish the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero.

Dynamics of stochastic FitzHugh–Nagumo systems with additive noise on unbounded thin domains

2019 ◽  
Vol 20 (03) ◽  
pp. 2050018
Author(s):  
Lin Shi ◽  
Dingshi Li ◽  
Xiliang Li ◽  
Xiaohu Wang
Keyword(s):  
Asymptotic Behavior ◽  
White Noise ◽  
Unbounded Domain ◽  
Existence And Uniqueness ◽  
Additive Noise ◽  
Upper Semicontinuity ◽  
Thin Domains ◽  
Additive White Noise ◽  
Random Attractors ◽  
Lower Dimensional

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain.

Wong–Zakai approximations of second-order stochastic lattice systems driven by additive white noise

2021 ◽  
pp. 2150050
Author(s):  
Yiju Chen ◽  
Chunxiao Guo ◽  
Xiaohu Wang
Keyword(s):  
White Noise ◽  
Additive Noise ◽  
Upper Semicontinuity ◽  
Second Order ◽  
Pullback Attractors ◽  
Lattice Systems ◽  
Additive White Noise ◽  
Random Attractors ◽  
Approximate System

In this paper, we study the Wong–Zakai approximations of a class of second-order stochastic lattice systems with additive noise. We first prove the existence of tempered pullback attractors for lattice systems driven by an approximation of the white noise. Then, we establish the upper semicontinuity of random attractors for the approximate system as the size of approximation approaches zero.

scholarly journals Weakly Nonlinear Magnetic Convection in a Nonuniformly Rotating Electrically Conductive Medium Under the Action of Modulation of External Fields

2020 ◽  
Keyword(s):  
Magnetic Field ◽  
Angular Velocity ◽  
Landau Equation ◽  
Temperature Modulation ◽  
External Fields ◽  
Ginzburg Landau ◽  
Electrically Conductive ◽  
Weakly Nonlinear ◽  
Conductive Fluid ◽  
Ginzburg Landau Equations

In this paper we studied the weakly nonlinear stage of stationary convective instability in a nonuniformly rotating layer of an electrically conductive fluid in an axial uniform magnetic field under the influence of: a) temperature modulation of the layer boundaries; b) gravitational modulation; c) modulation of the magnetic field; d) modulation of the angular velocity of rotation. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number nonlinear non-autonomous Ginzburg-Landau equations for the above types of modulation were obtaned. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various types of modulation of external fields and for different profiles of the angular velocity of the rotation of electrically conductive fluid.

Lubricated pipelining: stability of core–annular flow. Part 4. Ginzburg–Landau equations

1991 ◽  
Vol 227 ◽  
pp. 587-615 ◽  
Author(s):  
Kangping Chen ◽  
Daniel D. Joseph
Keyword(s):  
Annular Flow ◽  
Landau Equation ◽  
Finite Amplitude ◽  
Ginzburg Landau ◽  
Neutral Curves ◽  
Critical Reynolds Numbers ◽  
Ginzburg Landau Equations ◽  
Flow Part ◽  
Annular Flows ◽  
Monochromatic Waves

Nonlinear stability of core-annular flow near points of the neutral curves at which perfect core-annular flow loses stability is studied using Ginzburg-Landau equations. Most of the core-annular flows are always unstable. Therefore the set of core-annular flows having critical Reynolds numbers is small, so that the set of flows for which our analysis applies is small. An efficient and accurate algorithm for computing all the coefficients of the Ginzburg-Landau equation is implemented. The nonlinear flows seen in the experiments do not appear to be modulations of monochromatic waves, and we see no evidence for soliton-like structures. We explore the bifurcation structure of finite-amplitude monochromatic waves at criticality. The bifurcation theory is consistent with observations in some of the flow cases to which it applies and is not inconsistent in the other cases.

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