Random Attractor Research Articles

Random attractor for second-order stochastic delay lattice sine-Gordon equation

In this paper, we prove the existence of random mathcal{D}-attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov ε-entropy for the random mathcal{D}-attractor.

Random perturbations of an eco-epidemiological model

<p style='text-indent:20px;'>We consider random perturbations of a general eco-epidemiological model. We prove the existence of a global random attractor, the persistence of susceptibles preys and provide conditions for the simultaneous extinction of infectives and predators. We also discuss the dynamics of the corresponding random epidemiological <inline-formula><tex-math id="M1">\begin{document}$ SI $\end{document}</tex-math></inline-formula> and predator-prey models. We obtain for this cases a global random attractor, prove the prevalence of susceptibles/preys and provide conditions for the extinctions of infectives/predators.</p>

Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays

<p style='text-indent:20px;'>We study asymptotically autonomous dynamics for non-autonom-ous stochastic 3D Brinkman-Forchheimer equations with general delays (containing variable delay and distributed delay). We first prove the existence of a pullback random attractor not only in the initial space but also in the regular space. We then prove that, under the topology of the regular space, the time-fibre of the pullback random attractor semi-converges to the random attractor of the autonomous stochastic equation as the time-parameter goes to minus infinity. The general delay force is assumed to be pointwise Lipschitz continuous only, which relaxes the uniform Lipschitz condition in the literature and includes more examples.</p>

Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise

<p style='text-indent:20px;'>The global well-posedness and long-time mean random dynamics are studied for a high-dimensional non-autonomous stochastic nonlinear lattice pseudo-parabolic equation with <i>locally</i> Lipschitz drift and diffusion terms. The existence and uniqueness of three different types of weak pullback mean random attractors as well as their relations are established for the mean random dynamical systems generated by the solution operators. This is the first paper to study the well-posedness and dynamics of the stochastic lattice pseudo-parabolic equation even when the nonlinear noise reduces to the linear one.</p>

Kolmogorov Entropy of Random Attractors for Stochastic Lattice Systems With White Noise

Based on the asymptotic behavior theory for solutions of dissipative stochastic lattice systems, the Kolmogorov entropy of random attractors in stochastic Klein-Gordon-Schrdinger lattice dynamic systems with white noise was studied with the element decomposition method and the topological properties of polyhedral sphere covering in the finite dimensional space, and an upper bound for the Kolmogorov entropy of random attractors was obtained.

Random attractors for stochastic discrete long wave-short wave resonance equations driven by fractional Brownian motions

<abstract><p>We study the dynamical behavior of the solutions of stochastic discrete long wave-short wave resonance equations driven by fractional Brownian motions with Hurst parameter $ H\in(\frac{1}{2}, 1) $. And then we prove that the random dynamical system has a unique random equilibrium, which constitutes a singleton sets random attractor.</p></abstract>

LIMITING DYNAMICS OF NON-AUTONOMOUS STOCHASTIC GINZBURG-LANDAU EQUATIONS ON THIN DOMAINS

We examine the limiting dynamics of a class of non-autonomous stochastic Ginzburg-Landau equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. The existence and uniqueness of tempered pullback random attractors are established for the stochastic Ginzburg-Landau systems defined on $ (n+1) $-dimensional narrow domain. In addition, the upper semicontinuity of these attractors is obtained when a family of $ (n+1) $-dimensional thin domains collapses onto an $ n $-dimensional domain.

A Family of the Random Attractors for a Class of Generalized Kirchhoff-Type Equations

A Family of the Random Attractors for a Class of Generalized Kirchhoff-Type Equations

EXISTENCE AND UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR NON-AUTONOMOUS STOCHASTIC PLATE EQUATIONS WITH MULTIPLICATIVE NOISE ON <inline-formula><tex-math id="M1">$ \mathbb{R}^N $</tex-math></inline-formula>

<abstract> Based on the abstract theory of random attractors of non-autonomous non-compact dynamical systems, we investigate existence and the upper semi-continuity of random attractors for the non-autonomous stochastic plate equations with multiplicative noise defined on the entire space $ \mathbb{R}^n $. We extend and improve the results of [<xref ref-type="bibr" rid="b42">42</xref>] not only from the additive white noise to the multiplicative white noise, but also from the time-independent of forcing term $ g(x) $ to the time-dependent forcing term $ g(x, t) $. </abstract>

Almost continuity of a pullback random attractor for the stochastic $g$-Navier–Stokes equation

Almost continuity of a pullback random attractor for the stochastic $g$-Navier–Stokes equation

Random Attraction in the TASEP Model

The totally asymmetric simple exclusion process (TASEP) is a basic model of statistical mechanics that has found numerous applications. We consider the case of TASEP with a finite chain where particles may enter from the left and leave to the right at prescribed rates. This model can be formulated as a Markov process with a finite number of states. Due to the irreducibility of the process, it is well-known that the probability distribution on the states is globally attracted to a unique equilibrium distribution. We extend this result to the more detailed level of individual trajectories. To do so we formulate TASEP as a random dynamical system. Our main result is that the trajectories from all possible initial conditions contract to each other yielding the existence of a random attractor that consists of a single trajectory almost surely. This implies that in the long run TASEP “filters out” any perturbation that changes the state of the particles along the chain.

The Family of Exponential Attractors and Random Attractors for a Class of Kirchhoff Equations

The Family of Exponential Attractors and Random Attractors for a Class of Kirchhoff Equations

Pullback random attractors of stochastic strongly damped wave equations with variable delays on unbounded domains

&lt;abstract&gt;&lt;p&gt;In this paper, we consider the asymptotic behavior of solutions to stochastic strongly damped wave equations with variable delays on unbounded domains, which is driven by both additive noise and deterministic non-autonomous forcing. We first establish a continuous cocycle for the equations. Then we prove asymptotic compactness of the cocycle by tail-estimates and a decomposition technique of solutions. Finally, we obtain the existence of a tempered pullback random attractor.&lt;/p&gt;&lt;/abstract&gt;

Uniform random attractors for 2D non-autonomous stochastic Navier-Stokes equations

In this paper, we first establish the existence of uniform random attractor for 2D stochastic Navier-Stokes equation in H with deterministic non-autonomous external force being normal in Lloc2(R;V′), which is the measurable minimal compact set and uniformly attracts bounded random set in H in the sense of pullback. We also show that uniform random attractor with respect to the deterministic non-autonomous functions belonging to some symbol space coincides with uniform random attractor with respect to the initial time. Then we show that the uniform random attractor for the equation under consideration has regularity property in V when deterministic non-autonomous external force being normal in Lloc2(R;H).

Dynamics for stochastic Fitzhugh–Nagumo systems with general multiplicative noise on thin domains

This paper is devoted to studying the dynamics for stochastic Fitzhugh–Nagumo equations with general multiplicative noise on thin domains, where the noise is described by a general stochastic process instead of the usual Wiener process. The measurability of the solution operator is proved, which yields to the measurability of the random attractor. We then show the existence and upper semi‐continuity of these attractors when a family of (n + 1)‐dimensional thin domains degenerates onto an n‐dimensional domain under the topology of p‐times Lebesgue space.

Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise

<p style='text-indent:20px;'>In this paper we prove that the stochastic Navier-Stokes equations with stable Lévy noise generate a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Lévy noise (finite dimensional). We also deduce the existence of a Feller Markov Invariant Measure.

Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

Asymptotic behavior of two-dimensional stochastic nematic liquid crystal flows with multiplicative noise

Under non-periodic boundary conditions, we consider the long-term behavior of stochastic two-dimensional (2D) nematic liquid crystal flows with multiplicative noise. The structure of the corresponding stochastic model is different from that of the deterministic model. Noise destroys the basic balance law for the nematic liquid crystal flows, so we cannot employ the standard argument to obtain uniform a priori estimates of the solutions to this stochastic model. To overcome this problem, we use logarithmic energy estimates and the Itô formula in a Banach space to obtain uniform estimates that improve the previous result where the orientation field grows exponentially w.r.t. time t. In order to study the existence of a random attractor, we need to show that the solution is a stochastic flow. However, this is not obvious because of the particularly multiplicative noise in the orientation field. To show the flow property of the orientation field, we construct linear stochastic partial differential equations where the solutions are stochastic flows. After determining the relationship between these linear equations and the orientation field equation, we prove that each component of the orientation field is indeed a stochastic flow. The global well-posedness of the stochastic 2D nematic liquid crystal flows is only established for weak solutions, so the existence of the random attractor should be demonstrated in the weak solution space. Thus, using the standard method, we need to derive uniform a priori estimates in the strong solution space. However, the standard method fails because of the ill-posedness of the strong solution. We show that by proving the compactness property of the stochastic flow and using the regularity of the solutions, we can construct a compact absorbing ball in the weak solution space, which implies the existence of the random attractor. Finally, we show the existence of the invariant measure for the corresponding stochastic model, which is the first result for the long-term behavior of stochastic nematic liquid crystals under Dirichlet boundary conditions for the velocity field and Neumann boundary conditions for the orientation field.

Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains

This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space $\mathbb {R}^n$. The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ is caused by the lack of compact Sobolev embeddings on $\mathbb {R}^n$, as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46–57, 2018).

Regular random attractors for non-autonomous stochastic evolution equations with time-varying delays on thin domains

This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction–diffusion equations with time-varying delays on thin domains. First, we prove the existence and uniqueness of the regular random attractor. Then, we prove the upper semicontinuity of the regular random attractors for the equations on a family of (n + 1)-dimensional thin domains that collapses onto an n-dimensional domain.