scholarly journals Finite fractal dimensions of random attractors for stochastic FitzHugh–Nagumo system with multiplicative white noise

2016 ◽  
Vol 441 (2) ◽  
pp. 648-667 ◽  
Author(s):  
Shengfan Zhou ◽  
Zhaojuan Wang
Keyword(s):  
White Noise ◽  
Fractal Dimensions ◽  
Random Attractors ◽  
Multiplicative White Noise

scholarly journals Random Attractors for Stochastic Three-Component Reversible Gray-Scott System on Infinite Lattices

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Anhui Gu ◽  
Zhaojuan Wang ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise ◽  
Infinite Lattices

We prove the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices.

scholarly journals Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains

2018 ◽  
Vol 16 (1) ◽  
pp. 862-884
Author(s):  
Xiaoyao Jia ◽  
Xiaoquan Ding ◽  
Juanjuan Gao
Keyword(s):  
Dynamical Systems ◽  
White Noise ◽  
Reaction Diffusion ◽  
Random Parameter ◽  
Reaction Diffusion Equations ◽  
Random Dynamical Systems ◽  
Diffusion Equations ◽  
Uhlenbeck Process ◽  
Random Attractors ◽  
Multiplicative White Noise

AbstractIn this paper we investigate the stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domain ℝn (n ≥ 2). We first transform the retarded reaction-diffusion equations into the deterministic reaction-diffusion equations with random parameter by Ornstein-Uhlenbeck process. Next, we show the original equations generate the random dynamical systems, and prove the existence of random attractors by conjugation relation between two random dynamical systems. In this process, we use the cut-off technique to obtain the pullback asymptotic compactness.

scholarly journals Existence and Upper Semicontinuity of Attractors for Nonautonomous Stochastic Sine-Gordon Lattice Systems with Random Coupled Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Zhaojuan Wang ◽  
Shengfan Zhou
Keyword(s):  
White Noise ◽  
Weighted Space ◽  
Upper Semicontinuity ◽  
Lattice Systems ◽  
Random Attractors ◽  
Multiplicative White Noise

We study nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise. We first consider the existence of random attractors in a weighted space for this system and then establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

scholarly journals Random Attractors for Stochastic Three-Component Reversible Gray-Scott System with Multiplicative White Noise

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Anhui Gu
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise

The paper is devoted to proving the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with multiplicative white noise.

scholarly journals Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

2016 ◽  
Vol 130 ◽  
pp. 255-278 ◽  
Author(s):  
Tomás Caraballo ◽  
Xiaoying Han ◽  
Björn Schmalfuss ◽  
José Valero
Keyword(s):  
Dynamical Systems ◽  
White Noise ◽  
Random Attractors ◽  
Multiplicative White Noise ◽  
Lattice Dynamical Systems

scholarly journals Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on $ {\mathbb{R}}^N $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guifen Liu ◽  
Wenqiang Zhao
Keyword(s):  
Differential Equation ◽  
Parabolic Equation ◽  
White Noise ◽  
Parabolic Equations ◽  
Growth Exponent ◽  
Linear Parabolic Equation ◽  
Approximation Technique ◽  
Linear Parabolic Equations ◽  
Random Attractors ◽  
Multiplicative White Noise

<p style='text-indent:20px;'>In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of <inline-formula><tex-math id="M2">\begin{document}$ L^2( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> space. We establish the Wong-Zakai approximations of solutions in <inline-formula><tex-math id="M3">\begin{document}$ L^l( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> for arbitrary <inline-formula><tex-math id="M4">\begin{document}$ l\geq q $\end{document}</tex-math></inline-formula> in the sense of upper semi-continuity of their random attractors, where <inline-formula><tex-math id="M5">\begin{document}$ q $\end{document}</tex-math></inline-formula> is the growth exponent of the nonlinearity. The <inline-formula><tex-math id="M6">\begin{document}$ L^l $\end{document}</tex-math></inline-formula>-pre-compactness of attractors is proved by using the truncation estimate in <inline-formula><tex-math id="M7">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> and the higher-order bound of solutions.</p>

scholarly journals Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kush Kinra ◽  
Manil T. Mohan
Keyword(s):  
White Noise ◽  
Random Perturbation ◽  
Three Dimensional ◽  
Random Attractor ◽  
Sufficient Evidence ◽  
Dimensional Torus ◽  
Random Attractors ◽  
Multiplicative White Noise ◽  
2D And 3D ◽  
Upper And Lower Semicontinuity

<p style='text-indent:20px;'>This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional torus (<inline-formula><tex-math id="M2">\begin{document}$ n = 2, 3 $\end{document}</tex-math></inline-formula>):</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{f}, \ \nabla\cdot\boldsymbol{u} = 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ r\geq1 $\end{document}</tex-math></inline-formula>. We prove that the global attractor of the above system is singleton under small forcing intensity (<inline-formula><tex-math id="M4">\begin{document}$ r\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ r\geq 3 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M9">\begin{document}$ r = n = 3 $\end{document}</tex-math></inline-formula>). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all <inline-formula><tex-math id="M10">\begin{document}$ 1\leq r&lt;\infty $\end{document}</tex-math></inline-formula>, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for <inline-formula><tex-math id="M11">\begin{document}$ 3\leq r&lt;\infty $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M12">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ r = 3 $\end{document}</tex-math></inline-formula>), when the coefficient of random perturbation converges to zero.</p>

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