Continuous Random Dynamical System Research Articles

Dynamics of a random Hopfield neural lattice model with adaptive synapses and delayed Hebbian learning

UDC 517.9 A Dong–Hopfield neural lattice model with random external forcing and delayed response to the evolution of interconnection weights is developed and studied. The interconnection weights evolve according to the Hebbian learning rule with a decay term and contribute to changes in the states after a short delay. The lattice system is first reformulated as a coupled functional-ordinary differential equation system on an appropriate product space. Then the solution of the system is shown to exist and be unique. Furthermore it is shown that the system of equations generates a continuous random dynamical system. Finally, the existence of random attractors for the random dynamical system generated by the Dong–Hopfield model is established.

Wong-Zakai type approximations of rough random dynamical systems by smooth noise

This paper is devoted to the smooth and stationary Wong-Zakai approximations for a class of rough differential equations driven by a geometric fractional Brownian rough path ω with Hurst index H∈(13,12]. We first construct the approximation ωδ of ω by probabilistic arguments, and then use the rough path theory to obtain the Wong-Zakai approximation for the solution on any finite interval. Finally, both the original system and the approximative system generate a continuous random dynamical systems φ and φδ. As a consequence of the Wong-Zakai approximation of the solution, φδ converges to φ as δ→0.

Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise

<p style='text-indent:20px;'>We mainly consider the existence of random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) defined on the space of infinite sequences with complex-valued components. Then we transfer the considered stochastic Schrödinger lattice system into a random system with random parameters and quasi-periodic forces without white noise through Ornstein-Uhlenbeck process, whose solutions generate a jointly continuous NRDS. Thirdly, we prove the existence of a uniform absorbing random set for this NRDS. Fourthly, we construct a bounded closed random set and verify the Lipschitz continuity of the NRDS on this random set, and next we decompose the difference between two solutions into a sum of two parts including some random variables with bounded expectation. Finally, we obtain the existence of a random uniform exponential attractor for the considered system.</p>

Asymptotic behavior of non-autonomous fractional stochastic <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian equations with colored noise and delay on <inline-formula><tex-math id="M2">$ \mathbb{R}^n $</tex-math></inline-formula>

This paper deals with the asymptotic behavior of the solutions to a class of non-autonomous fractional stochastic $ p $-Laplacian equations with delay driven by nonlinear colored noise on the entire space $ \mathbb{R}^n $. We firstly considered the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the time delay. We then showed pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains.

Random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice systems in weighted space

<abstract><p>We mainly study the existence of random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice system with multiplicative white noise and quasi-periodic forces in weighted spaces. Firstly, the stochastic Schrödinger system is transformed into a random system without white noise by the Ornstein-Uhlenbeck process, whose solution generates a jointly continuous non-autonomous random dynamical system $ \Phi $. Secondly, we prove the existence of a uniform absorbing random set for $ \Phi $ in weighted spaces. Finally, we obtain the existence of a random uniform exponential attractor for the considered system $ \Phi $ in weighted space.</p></abstract>

Random uniform exponential attractors for non-autonomous stochastic discrete long wave-short wave resonance equations

In this paper, we study the dynamical behavior of the non-autonomous stochastic discrete long wave-short wave resonance system with multiplicative white noise and quasi-periodic forces. Firstly, we prove the uniqueness and existence for the solution of the system, which generates a jointly continuous random dynamical system $ \Phi $. Secondly, we prove the existence of a uniform absorbing random set for $ \Phi $ and estimate tail of the solution. Thirdly, we prove the Lipschitz continuity of the skew-product cocycle $ \pi $ defined on an extended space. Fourthly, we prove that the expectation of some random variables are bounded. Finally, we prove the existence of a random uniform exponential attractor for the considered system.

Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains

<abstract><p>This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations.</p></abstract>

On entropy and pressure via preimage structure for random dynamical systems

In this paper, entropy, pressure and equilibrium states are investigated for noninvertible random dynamical systems via the preimage structure. For a continuous random dynamical system F on a compact metric space and an F-invariant measure μ, the folding entropy Fμ(F) and the preimage entropy hm,μ(F) are introduced, and it is shown that these two measure-theoretic versions of entropies coincide with each other if F has uniform separation of preimages. Some fundamental properties for hm,μ(F) such as Shannon-McMillan-Breiman Theorem, Kolmogorov-Sinai Theorem and upper semi-continuity are established. From the topological viewpoint, the preimage pressures Pm(F,⋅), Pp(F,⋅) and Pp˜(F,⋅) (resp. topological preimage entropy hm(F), hp(F) and hp˜(F)) are introduced via the preimage structure of single points. Some basic properties are discussed, and particularly, variational principles are obtained. For a C1 nondegenerate random partially hyperbolic dynamical system F on a closed Riemannian manifold, we propose the notions of metric stable entropy hm,μs(F) and stable pressures Pms(F,⋅) and Pps(F,⋅) (resp. topological stable entropies hms(F) and hps(F)) via the preimage structure of local stable manifolds. Several basic properties are obtained and variational principles are established particularly. At the end, as an application of variational principles, p-equilibrium states (resp. s-equilibrium states) and p-tangent functionals (resp. s-tangent functionals) are studied.

On the induced measure-theoretic entropy for random dynamical systems

In this paper, we introduce the concepts of induced topological entropy and induced measure-theoretic entropy for random dynamical systems. By providing the topological version of induced measure-theoretic entropy, we establish the Katok entropy formula for a continuous bundle random dynamical system over an invertible ergodic system. We also consider the induced version with lower and upper pointwise dimensions, and establish a measure-theoretic entropy formula for lower and upper induced pointwise dimensions for a continuous bundle random dynamical system over an invertible ergodic system.

Random dynamical systems with positive entropy imply second type of distributional chaos

In this paper, we investigate the complexity of random dynamical systems. Assume that ϕ is a continuous random dynamical system on Polish space X over an ergodic Polish system (Ω,F,P,θ). Let K be a ϕ-invariant random set with compact ω-section K(ω) such that ϕ(ω)(K(ω))=K(θω). We proved that if htop(ϕ,K)>0, then there exists chaotic phenomenon (strong second type of distributional chaos) in this random dynamical system.

Existence and Upper Semi-Continuity of Random Attractors for Nonclassical Diffusion Equation with Multiplicative Noise on R<sup>n</sup>

This paper is concerned with the existence and upper semi-continuity of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity and multiplicative noise in H1(Rn). First, we study the existence and uniqueness of solutions by a noise arising in a continuous random dynamical system and the asymptotic compactness is established by using uniform tail estimate technique, and then the existence of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity. As a motivation of our results, we prove an existence and upper semi-continuity of random attractors with respect to the nonlinearity that enters the system together with the noise.

Random attractor and exponentially stability for the second order nonautonomous retarded lattice systems with multiplicative white noise

In this paper, we first present some conditions for the existence of a random attractor of continuous non-autonomous random dynamical systems (cocycle) and locally coupled stochastic retarded lattice dynamical systems in a space of continuous functions from finite closed interval into a product space of finite weighted spaces of infinite sequences. Then we consider the existence, singleton and exponentially stability of the random attractor for the second order non-autonomous retarded lattice systems with multiplicative white noise.

On a rough perturbation of the Navier–Stokes system and its vorticity formulation

We introduce a rough perturbation of the Navier–Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of strong solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong–Zakai result for Brownian driving paths, and show that for a large class of driving signals, the system generates a continuous random dynamical system. In dimension three, the noise is not enstrophy balanced, and we establish the existence of local in time solutions.

Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise

<p style='text-indent:20px;'>This paper is concerned with the pullback random attractors of nonautonomous nonlocal fractional stochastic <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian equation with delay driven by multiplicative white noise defined on bounded domain. We first prove the existence of a continuous nonautonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise. We then show pullback asymptotical compactness of solutions and the existence of tempered random attractors by utilizing the Arzela-Ascoli theorem and appropriate uniform estimates of the solutions. Finally, we establish the upper semicontinuity of the random attractors when time delay approaches zero.</p>

Random pullback and uniform exponential attractors for stochastic non-autonomous Navier-Stokes equations

We first prove the existence of a random pullback exponential attractor for the 2D Navier-Stokes equation with additive noise and non-autonomous forcing. Then, we introduce the definition and existence conditions of a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system, and prove the existence of a random uniform exponential attractor for the 2D Navier-Stokes equation with additive noise and quasi-periodic external forces.

Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay

This paper is concerned with the asymptotic behavior of the solutions to a class of non-autonomous nonlocal fractional stochastic parabolic equations with delay defined on bounded domain. We first prove the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise intensity. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform estimates of solutions in fractional Sobolev space \begin{document}$ H^\alpha(\mathbb{R}^n) $\end{document} with \begin{document}$ \alpha\in (0,1) $\end{document} as well as their time derivatives in \begin{document}$ L^2(\mathbb{R}^n) $\end{document} . Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively.

Uniform attractors for a class of stochastic evolution equations with multiplicative fractional noise

This paper studies the (random) uniform attractor for a class of non-autonomous stochastic evolution equations driven by a time-periodic forcing and multiplicative fractional noise with Hurst parameter bigger than 1/2. We first establish the existence and uniqueness results for the solution to the considered equation and show that the solution generates a jointly continuous non-autonomous random dynamical system (NRDS). Moreover, we prove the existence of the uniform attractor for this NRDS through stopping time technique. Particularly, a compact uniformly absorbing set is constructed under a smallness condition imposed on the fractional noise.

Random Attractors for 2D Stochastic Hydrodynamical-Type Systems

We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive noise that covers numerous 2D hydrodynamical models, such as the 2D Navier–Stokes equations, 2D Boussinesq equations, 2D MHD equations, etc., and also some 3D models, like the 3D Leray 𝛼-model. We prove the existence of random attractors for the associated continuous random dynamical systems. We also establish the upper semicontinuity of the random attractors as the parameter tends to zero.

Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh–Nagumo lattice systems with quasi-periodic forces and multiplicative noise

We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system defined on the product space of [Formula: see text]-weighted spaces of infinite sequences. Then, based on this criterion, we prove the existence of random uniform exponential attractors for stochastic lattice systems and stochastic FitzHugh–Nagumo lattice systems that are both with quasi-periodic forces.

Long Term Behavior for a Class of Stochastic Delay Lattice Systems inXρSpace

In this paper, we focus on the asymptotic behavior of solutions to stochastic delay lattice equations with additive noise and deterministic forcing. We first show the existence of a continuous random dynamical system for the equations. Then we investigate the pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractor inXρspace. Finally, ergodicity of the systems is achieved.