Attractors for Hopfield lattice model in weighted spaces

The investigation into the dynamic behavior of infinite lattice systems holds paramount significance in the realm of physical phenomena, particularly in mechanics. This intricate domain has captivated the attention of both mathematicians and physicists. In acknowledgment of the inherent noise prevalent in real-world environments, our study embraces this aspect by introducing a random term into our model. This deliberate inclusion of stochasticity engenders a novel perspective, giving rise to a stochastic lattice differential equation. This model proves to be a versatile tool for accurately characterizing spatial structures characterized by discrete components and the associated uncertainties that pervade them. This research elucidates the intricate interplay between lattice dynamics and environmental noise, shedding light on the complex behavior of such systems in a realistic context. Our result generalizes many results in three directions: extending the connections between the terms to non-linear, extending the connection neighborhood from 3 (as in most cases) to arbitrary value n, and extending the results that are in ℓ2 to ℓρ2.

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

Attractors for 2D quasi-geostrophic equations with and without colored noise in W2α−,p(ℝ2)

The asymptotic behavior of stochastic modified quasi-geostrophic equations with damping driven by colored noise is analyzed. In fact, the existence of random attractors is established in [Formula: see text] In particular, we prove also the existence of a global compact attractor for autonomous quasi-geostrophic equations with damping in [Formula: see text] Here, we do not add any modifying factor on the nonlinear term.

Dynamic behavior of stochastic p-Laplacian-type lattice equations

In this paper, we consider the dynamic behavior of stochastic [Formula: see text]-Laplacian-type lattice equations perturbed by a multiplicative noise. Under weaker dissipative conditions compared to the cases of stochastic [Formula: see text]-Laplacian-type equations in bounded and unbounded domains, we first obtain the existence of a unique random attractor. We also establish the approximation of the random attractors from finite lattice to infinite lattice, which indicates that the family of random attractors is upper and lower semi-continuous when the number of the lattice nodes tends to infinity.

Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains

The existence of a unique random attractors in [Formula: see text] for a stochastic reaction-diffusion equation with time-dependent external forces is proved. Due to the presence of both random and non-autonomous deterministic terms, we use a new theory of random attractors which is introduced in [B. Wang, J. Differential Equations 253 (2012) 1544–1583] instead of the usual one. The asymptotic compactness of solutions in [Formula: see text] is established by combining “tail estimate” technique and some new estimates on solutions. This work improves some recent results about the regularity of random attractors for stochastic reaction-diffusion equations.

Eluding catastrophic shifts

Transitions between regimes with radically different properties are ubiquitous in nature. Such transitions can occur either smoothly or in an abrupt and catastrophic fashion. Important examples of the latter can be found in ecology, climate sciences, and economics, to name a few, where regime shifts have catastrophic consequences that are mostly irreversible (e.g., desertification, coral reef collapses, and market crashes). Predicting and preventing these abrupt transitions remains a challenging and important task. Usually, simple deterministic equations are used to model and rationalize these complex situations. However, stochastic effects might have a profound effect. Here we use 1D and 2D spatially explicit models to show that intrinsic (demographic) stochasticity can alter deterministic predictions dramatically, especially in the presence of other realistic features such as limited mobility or spatial heterogeneity. In particular, these ingredients can alter the possibility of catastrophic shifts by giving rise to much smoother and easily reversible continuous ones. The ideas presented here can help further understand catastrophic shifts and contribute to the discussion about the possibility of preventing such shifts to minimize their disruptive ecological, economic, and societal consequences.

Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms

We prove the existence and uniqueness of tempered random attractors for stochastic Reaction–Diffusion equations on unbounded domains with multiplicative noise and deterministic non-autonomous forcing. We establish the periodicity of the tempered attractors when the stochastic equations are forced by periodic functions. We further prove the upper semicontinuity of these attractors when the intensity of stochastic perturbations approaches zero.

RANDOM ATTRACTORS OF STOCHASTIC NON-NEWTONIAN FLUIDS ON UNBOUNDED DOMAIN

We consider the stochastic non-Newtonian fluids defined on a two-dimensional Poincaré unbounded domain, and prove that it generates an asymptotically compact random dynamical system. Then, we establish the existence of random attractor for the corresponding random dynamical system. Random attractor is invariant and attracts every pullback tempered random set.

ASYMPTOTIC BEHAVIOR OF STOCHASTIC DISSIPATIVE QUANTUM ZAKHAROV EQUATIONS

The stochastic dissipative quantum Zakharov equations with white noise are studied. The existence and uniqueness of solutions are obtained by using the standard Galerkin approximation method on the basis of the time uniform a priori estimates in various spaces. Moreover, the asymptotic behaviors of the solutions for the stochastic dissipative quantum Zakharov equations with white noise are also investigated.

RANDOM ATTRACTORS FOR SECOND ORDER STOCHASTIC LATTICE DYNAMICAL SYSTEMS WITH MULTIPLICATIVE NOISE IN WEIGHTED SPACES

In this paper, we study the asymptotic behavior of solutions of a second order stochastic lattice differential equation with multiplicative white noise in weighted spaces. We first provide some sufficient conditions for the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.

ASYMPTOTIC COMPACTNESS OF STOCHASTIC COMPLEX GINZBURG–LANDAU EQUATION ON AN UNBOUNDED DOMAIN

The Ginzburg–Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. In this paper, we consider the complex Ginzburg–Landau (CGL) equations on the whole real line perturbed by an additive spacetime white noise. Our main result shows that it generates an asymptotically compact stochastic or random dynamical system. This is a crucial property for the existence of a stochastic attractor for such CGL equations. We rely on suitable spaces with weights, due to the regularity properties of spacetime white noise, which gives rise to solutions that are unbounded in space.