Fractal Dimension Of Random Attractors Research Articles

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

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.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

We prove some conditions for bounding the fractal dimension of random invariant sets of non-autonomous random dynamical systems on separable Banach spaces. Then we apply these conditions to prove the finiteness of fractal dimension of random attractor for stochastic 2D hydrodynamical type equations with linear additive white noise in bounded domains or unbounded domains satisfying the Poincaré inequality.

Fractal Dimension of Random Attractor for a Stochastic Lattice System with White Noise

Fractal Dimension of Random Attractor for a Stochastic Lattice System with White Noise

Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise

<p style='text-indent:20px;'>In this paper, we mainly consider the existence of random attractor and random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. First step, the well-posedness and the existence of a random attractor for the cocycle associated with the considered system is established. Second step, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero. Third step, we prove the regularity of random attractor in a higher regular space by the "iteration" method. Finally, we give the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor.</p>

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

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.

Finite fractal dimension of random attractors for non-autonomous fractional stochastic reaction–diffusion equations in ℝ

This paper investigates the asymptotic behavior of solutions for non-autonomous fractional stochastic reaction–diffusion equations with multiplicative noise in . We prove the existence and uniqueness of the tempered pullback random attractor for the equations in and obtain the finite fractal dimension for the pullback random attractor. Two main difficulties here are that the fractional Laplacian operator is non-local and the Sobolev embedding is not compact on unbounded domains. To solve this, we derive the tail-estimates of solutions of the equation and decompose the solutions into a sum of three parts, which one part is finite-dimensional and other two parts are quickly decay in mean sense.

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg—Landau Equations

This paper considers the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with α e (0,1). First, we give some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Second, we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equation in H. At last, we prove the finiteness of fractal dimension of random attractors.

Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise

We mainly consider the existence of a random exponential attractor (positive invariant compact measurable set with finite fractal dimension and attracting orbits exponentially) for stochastic discrete long wave-short wave resonance equation driven by multiplicative white noise. Firstly, we prove the existence of a random attractor of the considered equation by proving the existence of a uniformly tempered pullback absorbing set and making an estimate on the 'tails' of solutions. Secondly, we show the Lipschitz property of the solution process generated by the considered equation. Finally, we prove the existence of a random exponential attractor of the considered equation, which implies the finiteness of fractal dimension of random attractor.

Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg–Landau equations with multiplicative noise

ABSTRACTThis paper deals with the asymptotic behaviour of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by multiplicative noise with . We first present some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Then we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equations in . At last, we prove the finiteness of fractal dimension of random attractors.

Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation

In this paper, we first prove the existence of a random attractor for stochastic non-autonomous strongly damped wave equations with additive white noise. Then we apply a criteria to obtain an upper bound of fractal dimension of the random attractor of considered system.

Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise

We mainly consider the existence of a random exponential attractor (positive invariant compact measurable set with finite fractal dimension and attracting orbits exponentially) for stochastic discrete long wave-short wave resonance equation driven by multiplicative white noise. Firstly, we prove the existence of a random attractor of the considered equation by proving the existence of a uniformly tempered pullback absorbing set and making an estimate on the 'tails' of solutions. Secondly, we show the Lipschitz property of the solution process generated by the considered equation. Finally, we prove the existence of a random exponential attractor of the considered equation, which implies the finiteness of fractal dimension of random attractor.

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

In this paper, we consider the asymptotic behavior of solutions for stochastic non-autonomous FitzHugh–Nagumo system with multiplicative white noise. First we prove the existence of random attractor of the random dynamical system generated by the solutions of considered system. Then we present some conditions for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable Banach space and apply these conditions to prove the finiteness of fractal dimension of random attractor.

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

In this paper, we first establish a criteria for estimating the upper bound of fractal dimension of a random compact invariant set of a non-autonomous random dynamical system on a separable Banach space and give some special conditions which can be easily verified to some stochastic partial differential equations for bounding the fractal dimension of random attractors. Then we apply these conditions to obtain an upper bound of fractal dimension of random attractor for stochastic damped wave equation with additive white noise.

Fractal dimension of random attractors for stochastic non-autonomous reaction–diffusion equations

In this paper, we first give some conditions for bounding the fractal dimension of a random invariant set for a non-autonomous random dynamical system on a separable Banach space. Then we apply these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic reaction–diffusion equations with multiplicative white noise and additive white noise.

Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise

In this paper, we first present some conditionsfor bounding the fractal dimension of a random invariant set of anon-autonomous random dynamical system on a separable Banach space. Then weapply these conditions to prove the finiteness of fractal dimension ofrandom attractor for stochastic damped wave equation with linear multiplicativewhite noise.