Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation

Zhou, Sizhong; Zhao, Min

doi:10.1016/j.na.2015.12.013

Cited by 21 publications

(28 citation statements)

References 23 publications

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“…where N ε (A 0 (τ, ω)) is the minimal number of balls with radius ε > 0 covering A 0 (τ, ω) in H. Now let us check that A 0 satisfies the conditions of (I) of Theorem 2.2 of [41] one by one. It is easy to see that A 0 satisfies contions (H1) and (H2) of Theorem 2.2 of [41]. First, we prove the Lipschitz property of Φ 0 on A 0 .…”

Section: 1mentioning

confidence: 84%

“…In the rest of this subsection, we focus to estimate an upper bound of the fractal dimension of A 0 for the continuous cocycle {Φ 0 (t, τ, ω)} t≥0,τ ∈R,ω∈Ω associated with (59) by (I) of Theorem 2.2 of [41].…”

Section: 1mentioning

confidence: 99%

“…Taking t = t 0 = 12 ln 2 dλn 0 +1 in (127), then by (I) of Theorem 2.2 of [41], for any τ ∈ R, ω ∈ Ω, we get (136).…”

Section: 1mentioning

confidence: 99%

“…Wang [30,31] established an efficacious theory about the existence and upper semi-continuity of random attractors for cocycles. Later, Zhou et al [40,41,42] established a criteria for bounding the fractal dimension of a random invariant set for a cocycle and applied to some stochastic partial differential equations.…”

mentioning

confidence: 99%

“…Motivated by [28,30,31,40,41], we will consider the existence, the uppersemicontinuity, the fractal dimensions of random attractors for the stochastic nonautonomous Boissonade systems (1) and (2).…”

mentioning

confidence: 99%

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Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises

Zhao¹,

Zhou²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we study the long-term dynamical behavior of stochastic Boissonade systems with time-dependent deterministic forces, additive white noise and multiplicative white noise. We first prove the existence of random attractor for the considered systems. And then we establish the upper semi-continuity of random attractors for the systems as the coefficient of quadratic term tends to zero and intensities of the noises approach zero, respectively. At last, we obtain an upper bound of fractal dimension of the random attractors for both systems without quadratic term.

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Section: 1mentioning

confidence: 84%

Section: 1mentioning

confidence: 99%

“…Taking t = t 0 = 12 ln 2 dλn 0 +1 in (127), then by (I) of Theorem 2.2 of [41], for any τ ∈ R, ω ∈ Ω, we get (136).…”

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises

Zhao¹,

Zhou²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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Fractal dimension of random attractors for nonautonomous stochastic strongly damped wave equations on ℝN

Li,

2024

Math Methods in App Sciences

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In this article, we investigate the dynamics of a nonautonomous stochastic strongly damped wave equation defined on . We first use the energy equation and tail‐estimates to prove the asymptotic compactness of the solutions and obtain the existence of a unique pullback random attractor for the equation with critical nonlinearity. Then, we give an upper bound of fractal dimension of the random attractor when the nonlinearity is of subcritical growth. The unboundedness of the physical space will impose difficulties on the estimation for the upper bound of fractal dimension of the random attractor.

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Finite-Dimensionality of Tempered Random Uniform Attractors

2021

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Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in $$L^2$$ L 2 is established by the squeezing approach and that in $$H_0^1$$ H 0 1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

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Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation

Cited by 21 publications

References 23 publications

Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises

Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises

Fractal dimension of random attractors for nonautonomous stochastic strongly damped wave equations on ℝN

Finite-Dimensionality of Tempered Random Uniform Attractors

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