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 Random attractors for stochastic delay wave equations on $ \mathbb{R}^n $ with linear memory and nonlinear damping

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jingyu Wang ◽  
Yejuan Wang ◽  
Lin Yang ◽  
Tomás Caraballo
Keyword(s):  
Wave Equation ◽  
White Noise ◽  
Unbounded Domain ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Nonlinear Damping ◽  
Random Attractor ◽  
Stochastic Delay ◽  
Additive White Noise ◽  
Random Attractors

<p style='text-indent:20px;'>A non-autonomous stochastic delay wave equation with linear memory and nonlinear damping driven by additive white noise is considered on the unbounded domain <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. We establish the existence and uniqueness of a random attractor <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> that is compact in <inline-formula><tex-math id="M3">\begin{document}$ C{([-h, 0];H^1(\mathbb{R}^n))}\times C{([-h, 0];L^2(\mathbb{R}^n))}\times L_\mu^2(\mathbb{R}^+;H^1(\mathbb{R}^n)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ 1\leqslant n \leqslant 3 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Asymptotic behavior of non-autonomous fractional p-Laplacian equations driven by additive noise on unbounded domains

2020 ◽  
pp. 2050020
Author(s):  
Renhai Wang ◽  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Additive Noise ◽  
Random Coefficients ◽  
Asymptotic Behavior Of Solutions ◽  
Uniqueness Of Solutions ◽  
Additive White Noise ◽  
Approach Zero ◽  
Drift Terms ◽  
Laplacian Equations

This paper deals with the asymptotic behavior of solutions to non-autonomous, fractional, stochastic [Formula: see text]-Laplacian equations driven by additive white noise and random terms defined on the unbounded domain [Formula: see text]. We first prove the existence and uniqueness of solutions for polynomial drift terms of arbitrary order. We then establish the existence and uniqueness of pullback random attractors for the system in [Formula: see text]. This attractor is further proved to be a bi-spatial [Formula: see text]-attractor for any [Formula: see text], which is compact, measurable in [Formula: see text] and attracts all random subsets of [Formula: see text] with respect to the norm of [Formula: see text]. Finally, we show the robustness of these attractors as the intensity of noise and the random coefficients approach zero. The idea of uniform tail-estimates as well as the method of higher-order estimates on difference of solutions are employed to derive the pullback asymptotic compactness of solutions in [Formula: see text] for [Formula: see text] in order to overcome the non-compactness of Sobolev embeddings on [Formula: see text] and the nonlinearity of the fractional [Formula: see text]-Laplace operator.

scholarly journals Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains

2019 ◽  
Vol 17 (1) ◽  
pp. 1281-1302 ◽  
Author(s):  
Xiaobin Yao ◽  
Xilan Liu
Keyword(s):  
Asymptotic Behavior ◽  
Additive Noise ◽  
Upper Semicontinuity ◽  
Unbounded Domains ◽  
Asymptotic Behavior Of Solutions ◽  
Plate Equation ◽  
Uniform Estimates ◽  
Estimates Of Solutions ◽  
Random Attractors

Abstract We study the asymptotic behavior of solutions to the non-autonomous stochastic plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence and upper semicontinuity of random attractors.

scholarly journals Fractal Dimension for the Nonautonomous Stochastic Fifth-Order Swift–Hohenberg Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yanfeng Guo ◽  
Chunxiao Guo ◽  
Yongping Xi
Keyword(s):  
Fractal Dimension ◽  
White Noise ◽  
Bounded Domain ◽  
Unbounded Domain ◽  
Random Attractor ◽  
Additive White Noise ◽  
Random Attractors ◽  
Fifth Order

Some dynamics behaviors for the nonautonomous stochastic fifth-order Swift–Hohenberg equation with additive white noise are considered. The existence of pullback random attractors for the nonautonomous stochastic fifth-order Swift–Hohenberg equation with some properties is mainly investigated on the bounded domain and unbounded domain, through the Ornstein–Uhlenbeck transformation and tail-term estimates. Furthermore, on the basis of some conditions, the finiteness of fractal dimension of random attractor is proved.

Random attractors of FitzHugh–Nagumo systems driven by colored noise on unbounded domains

2019 ◽  
Vol 19 (05) ◽  
pp. 1950035
Author(s):  
Anhui Gu ◽  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
White Noise ◽  
Correlation Time ◽  
Existence And Uniqueness ◽  
Colored Noise ◽  
Stochastic Systems ◽  
Unbounded Domains ◽  
Limiting Behavior ◽  
Uniform Estimates ◽  
Random Attractors

We investigate the pathwise asymptotic behavior of the FitzHugh–Nagumo systems defined on unbounded domains driven by nonlinear colored noise. We prove the existence and uniqueness of tempered pullback random attractors of the systems with polynomial diffusion terms. The pullback asymptotic compactness of solutions is obtained by the uniform estimates on the tails of solutions outside a bounded domain. We also examine the limiting behavior of the FitzHugh–Nagumo systems driven by linear colored noise as the correlation time of the colored noise approaches zero. In this respect, we prove that the solutions and the pullback random attractors of the systems driven by linear colored noise converge to that of the corresponding stochastic systems driven by linear white noise.

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.

RANDOM ATTRACTOR FOR THE 3D VISCOUS STOCHASTIC PRIMITIVE EQUATIONS WITH ADDITIVE NOISE

2009 ◽  
Vol 09 (02) ◽  
pp. 293-313 ◽  
Author(s):  
HONGJUN GAO ◽  
CHENGFENG SUN
Keyword(s):  
White Noise ◽  
Existence And Uniqueness ◽  
Additive Noise ◽  
Strong Solutions ◽  
Random Attractor ◽  
Primitive Equations ◽  
Additive White Noise

In this article, we obtain the existence and uniqueness of strong solutions to 3D viscous stochastic primitive equations (PEs) and the random attractor for 3D viscous PEs with additive white noise.

scholarly journals Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Colored Noise ◽  
Unbounded Domains ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Natural Energy ◽  
Random Attractors ◽  
Supercritical Nonlinearity

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\le 6 $\end{document}</tex-math></inline-formula>. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.</p>

Effect of White Noise and Dichotomous Noise in a Bistable System

2011 ◽  
Vol 295-297 ◽  
pp. 2143-2146 ◽  
Author(s):  
Feng Guo ◽  
Xiao Feng Cheng ◽  
Xiao Dong Yuan ◽  
Shao Bo He
Keyword(s):  
White Noise ◽  
Adiabatic Approximation ◽  
Additive Noise ◽  
Signal To Noise Ratio ◽  
Bistable System ◽  
Signal To Noise ◽  
Dichotomous Noise ◽  
System Parameters ◽  
Additive White Noise ◽  
Asymmetric Dichotomous Noise

The stochastic resonance in a bistable system subject to asymmetric dichotomous noise and multiplicative and additive white noise is investigated. By using the properties of the dichotomous noise, under the adiabatic approximation condition, the expression of the signal-to-noise ratio (SNR) is obtained. It is found that the SNR is a non-monotonic function of the asymmetry of the dichotomous noise, and it varies non-monotonously with the intensities of the multiplicative and additive noise as well as with the system parameters. Moreover, the SNR depends on the correlation rate of the dichotomous noise.

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