Stochastic FitzHugh-Nagumo System Research Articles

Bi-spatial random attractor for stochastic FitzHugh-Nagumo systems on unbounded thin domain

Bi-spatial random attractor for stochastic FitzHugh-Nagumo systems on unbounded thin domain

Invariant measures for stochastic FitzHugh-Nagumo delay lattice systems with long-range interactions in weighted space

<abstract><p>The focus of this paper lies in exploring the limiting dynamics of stochastic FitzHugh-Nagumo delay lattice systems with long-range interactions and nonlinear noise in weighted space. To begin, we established the well-posedness of solutions to these stochastic delay lattice systems and subsequently proved the existence and uniqueness of invariant measures.</p></abstract>

Dynamics of stochastic FitzHugh–Nagumo system on unbounded domains with memory

In this paper, we consider the non-autonomous stochastic FitzHugh–Nagumo system with hereditary memory and a very large class of nonlinearities, which has no restriction on the upper growth of the nonlinearity. The existence of a random pullback attractor is established for this system in all N-dimensional space.

Backward regularity of attractors for lattice FitzHugh-Nagumo system with double random coefficients

In this paper, we investigate the backward compact dynamical behavior for the stochastic lattice FitzHugh-Nagumo system with double random coefficients and multiplicative noise. It is proved, under some suitable conditions on the nonlinear term and the body force, that the system has a unique backward compact bi-spatial random (ℓσ2×ℓσ2,ℓσp×ℓσ2)-attractor for any p>2. By such a bi-spatial attractor we mean an invariant backward compact set in ℓσp×ℓσ2 that pullback attracts all nonempty subsets of ℓσ2×ℓσ2 under another topology of ℓσp×ℓσ2. The time-dependent uniform pullback asymptotical compactness of the solution operators in ℓσp×ℓσ2 is proved by virtue of a cut-off technique and considering two different attracting universes to overcome several difficulties caused by the lack of compact Sobolev embeddings in the case of a lattice system and the unknown measurability of random attractor.

Wong-Zakai approximations and attractors for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains

In this paper, we study the Wong-Zakai approximations and long term behavior of the stochastic FitzHugh-Nagumo system driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation system. Then we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic FitzHugh-Nagumo system for both additive and multiplicative noise on unbounded domains. The tail estimates and decomposition method are employed to derive the pullback asymptotic compactness of solutions in order to overcome the obstacles caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the absence of regularity of one component of the solutions.

Wong–Zakai approximations of the non-autonomous stochastic FitzHugh–Nagumo system on RN in higher regular spaces

In this paper, we consider the Wong–Zakai approximations of a non-autonomous stochastic FitzHugh–Nagumo system driven by a multiplicative white noise with an arbitrary intensity. The convergence of solutions of the path-wise deterministic system to that of the corresponding stochastic system is established in higher regular spaces by means of a new iteration technique and an optimal multiplier at different stages. Furthermore, we prove that the random attractor of the path-wise deterministic system converges to that of the non-autonomous stochastic FitzHugh–Nagumo system in higher regular spaces when the size of approximation vanishes, with much looser conditions on the nonlinearity.

Wong–Zakai approximations and long term behavior of stochastic FitzHugh–Nagumo system

This paper is concerned with the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of stochastic FitzHugh–Nagumo system driven by white noise. We prove the existence and uniqueness of pullback random attractors for the approximate system under much weaker conditions than the original system. When the system is driven by additive white noise, we also prove the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.

Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains

The regularity of random attractors is considered for the non-autonomous fractional stochastic FitzHugh-Nagumo system. We prove that the system has a pullback random attractor that is compact in Hs(ℝn) × L2(ℝn) and attracts all tempered random sets of L2(ℝn) × L2(ℝn) in the topology of Hs(ℝn) × L2(ℝn) with s ∈ (0, 1). By the idea of positive and negative truncations, spectral decomposition in bounded domains, and tail estimates, we achieved the desired results.

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 uniform attractor and random cocycle attractor for non-autonomous stochastic FitzHugh–Nagumo system on unbounded domains

In this paper, we introduce the concept of uniform pullback asymptotic compactness of a non-autonomous random dynamical system, and prove the existence of random uniform and cocycle attractor with autonomous attraction universes for non-autonomous stochastic FitzHugh–Nagumo system with multiplicative noise defined on whole space. Some properties of random uniform and cocycle attractor are shown. The method of tail estimates plays an important role.

Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on ℝN driven by nonlinear noise

This article is concerned with the well-posedness as well as long-term dynamics of a wide class of non-autonomous, non-local, fractional, stochastic FitzHugh-Nagumo systems driven by nonlinear noise defined on the entire spaceℝRN. The well-posedness is proved for the systems with polynomial drift terms of arbitrary order as well as locally Lipschitz nonlinear diffusion terms by utilizing the pathwise and mean square uniform estimates. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. The existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion terms. The idea of uniform tail-estimates of the solutions in the appropriate spaces is employed to derive the tightness of a family of probability distributions of the solutions in order to overcome the non-compactness of the standard Sobolev embeddings on ℝN as well as the lack of smoothing effect on one component of the solutions. The results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian.

Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains

<p style='text-indent:20px;'>This paper is devoted to bi-spatial random attractors of the stochastic Fitzhugh-Nagumo equations with additive noise on thin domains when the terminate space is the Sobolev space. We first established the existence of random attractor on regular space and then show that the upper semi-continuity of these attractors under the Sobolev norm when a family of <inline-formula><tex-math id="M1">\begin{document}$ (n+1) $\end{document}</tex-math></inline-formula>-dimensional thin domains degenerates onto an <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional domain.

Pullback Random Attractors for Non-Autonomous Stochastic Fractional FitzHugh-Nagumo System

This paper is concerned with the asymptotic behavior of solutions for a class of non-autonomous fractional FitzHugh-Nagumo equations deriven by additive white noise. We first provide some sufficient conditions for the existence and uniqueness of solutions, and then prove the existence and uniqueness of tempered pullback random attractors for the random dynamical system generated by the solutions of considered equations in an appropriate Hilbert space. The proof is based on the uniform estimates and the decomposition of dynamical system.

Dynamics of stochastic FitzHugh–Nagumo systems with additive noise on unbounded thin domains

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain.

Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains

This article is concerned with the asymptotic behavior of solutions of a class of non-autonomous stochastic Fitzhugh-Nagumo systems driven by multiplicative white noise on unbounded thin domains. The existence and uniqueness of tempered pullback random attractors are proved for the systems defined on (n+1)−dimensional unbounded narrow domains. We also establish the upper semicontinuity of these attractors when a family of (n+1)−dimensional unbounded thin domains collapses onto an n−dimensional unbounded domain. A cut-off method and a decomposition technique are employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the lack of regularity of one component of the solutions.

Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises

In this article, we study the dynamical behaviour of solutions of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ with both multiplicative noises and non-autonomous forces, where the nonlinearity is a polynomial-like growth function of arbitrary order. An asymptotic smoothing effect of this system is demonstrated, namely, that the random pullback attractor in the initial space $L^2(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$ is actually a compact, measurable and attracting set in $H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$. A difference estimates method, rather than the usual truncation estimate and spectrum decomposition technique, is employed to overcome the lack of Sobolev compact embedding in $H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$, despite of the loss of the high-order integrability of the difference of solutions for this system.

Continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system on [formula omitted

In this article, we use the so-called difference estimate method to investigate the continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system with a general nonlinearity. Firstly, under weak assumptions on the noise coefficient, we prove the existence of a pullback attractor in L2(RN)×L2(RN) by using the tail estimate method and a certain compact embedding on bounded domains. Secondly, although the difference of the first component of solutions possesses at most p-times integrability where p is the growth exponent of the nonlinearity, we overcome the absence of higher-order integrability and establish the continuity of solutions in (Lp(RN)∩H1(RN))×L2(RN) with respect to the initial values belonging to L2(RN)×L2(RN). As an application of the result on the continuity, the existence of a pullback attractor in (Lp(RN)∩H1(RN))×L2(RN) is proved for arbitrary N≥1 and p>2.

REGULARITY OF PULLBACK ATTRACTORS AND RANDOM EQUILIBRIUM FOR NON-AUTONOMOUS STOCHASTIC FITZHUGH-NAGUMO SYSTEM ON UNBOUNDED DOMAINS

This paper is concerned with the stochastic Fitzhugh-Nagumo system with non-autonomous terms as well as Wiener type multiplicative noises. By using the so-called notions of uniform absorption and uniformly pullback asymptotic compactness, the existences and upper semi-continuity of pullback attractors are proved for the generated random cocycle in Ll(RN)×L2(RN) for any l ∈ (2,p]. The asymptotic compactness of the first component of the system in Lp(RN) is proved by a new asymptotic a priori estimate technique, by which the plus or minus sign of the nonlinearity at large values is not required. Moreover, the condition on the existence of the unique random fixed point is obtained, in which case the influence of physical parameters on the attractors is analysed.

Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems in weighted spaces

In this paper, we first consider the existence of random attractors for the non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and additive white noise in a weighted space $l_{\rho }^{2} \times l_{\rho }^{2} $ ; then we establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Strong averaging principle for two-time-scale non-autonomous stochastic FitzHugh-Nagumo system with jumps

In this paper, we study an averaging principle for stochastic FitzHugh-Nagumo system with different time scales driven by cylindrical Wiener processes and Poisson jumps, where the slow equation is non-autonomous and the fast equation is autonomous case. Under suitable assumptions, we show that the slow component mean-square strongly converges to the solution of the corresponding averaging equation, and the rate of the convergence as a by-product is also affirmed. Finally, we give some open problems which are derived from this paper.