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

Zhao, Wenqiang

doi:10.3934/dcdsb.2018251

Cited by 6 publications

(6 citation statements)

References 45 publications

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“…Recently, properties and optimal control of stochastic FHN (SFHN) models are extensively studied only in the finite dimensional space (see the references [10][11][12][13][14][15][16] ). In specific, the solvability and optimal control of SFHN model is studied by Barbu et al 17 ; optimal control of SFHN model with nonlinear diffusion term is studied by Cordoni and Di Persio 18 via rescaling transformation, Ekeland's variational principle, and big-bang optimal control theory.…”

Section: Introductionmentioning

confidence: 99%

Non‐instantaneous impulsive stochastic FitzHugh–Nagumo equation with fractional Brownian motion

Durga

Malik

2023

Math Methods in App Sciences

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This article is devoted to examine the exponential behavior of stochastic FitzHugh-Nagumo equation in Hilbert space. Initially, the proposed model is reformulated into an abstract space, and the existence of mild solution is constructed by employing Mönch fixed-point theorem and Hausdorff measure of non-compactness. Furthermore, suitable conditions are developed to prove the proposed model's exponential behavior which reflects the stable generation and transmission of electric signal in neurons. At last, an example is presented to verify the established theoretical concepts.

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Section: Introductionmentioning

confidence: 99%

Non‐instantaneous impulsive stochastic FitzHugh–Nagumo equation with fractional Brownian motion

Durga

Malik

2023

Math Methods in App Sciences

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“…The proof was based on an abstract invariant manifold theorem for dynamical systems on a Banach space. Note that in [21][22][23], the spectral gap condition may fail and the so-called spatial averaging principle (PSA) is used to construct the manifolds. Now, let us return to (1.1).…”

Section: Introductionmentioning

confidence: 99%

Global finite‐dimensional invariant manifolds of the Boissonade system

Liu

2023

Math Methods in App Sciences

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We consider the long‐time global dynamics of the Boissonade system. This system is a activator–inhibitor model that exhibits Turing structures for describing the relation between the genuine homogeneous 2D systems and the 3D monolayers. Assuming large diffusivity of the activator, we prove the existence of a finite‐dimensional manifold of global type by using a spatial averaging principle of local type and uniform dissipativity estimates. The manifold is locally invariant under the semiflow, which attracts uniformly exponentially those solutions with initial values having a certain regularity but attracts uniformly those solutions starting from the phase space.

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“…The continuity of random attractors includes both upper and lower semi-continuities. The upper semi-continuity describes the nonexplosive phenomenon, which is expected to hold widely, see the abstract results in [5,29,31] and many applications in [7,16,17,18,20,35,38,42,43]. The lower semi-continuity describes the non-implosive phenomenon, which is extremely hard to prove in partial differential equations (PDE) as pointed out by Carvalho et al [6, page 65].…”

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confidence: 99%

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

Yang

Long

2022

DCDS-B

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<p style='text-indent:20px;'>We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset <inline-formula><tex-math id="M1">\begin{document}$ \Lambda^* $\end{document}</tex-math></inline-formula> of the parameterized space such that the binary map <inline-formula><tex-math id="M2">\begin{document}$ (\lambda, s)\mapsto A_\lambda(\theta_s \omega) $\end{document}</tex-math></inline-formula> is continuous at all points of <inline-formula><tex-math id="M3">\begin{document}$ \Lambda^*\times \mathbb{R} $\end{document}</tex-math></inline-formula> with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space <inline-formula><tex-math id="M4">\begin{document}$ (0, \infty] $\end{document}</tex-math></inline-formula> and the infinity of noise means that the equation is deterministic.</p>

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

Cited by 6 publications

References 45 publications

Non‐instantaneous impulsive stochastic FitzHugh–Nagumo equation with fractional Brownian motion

Non‐instantaneous impulsive stochastic FitzHugh–Nagumo equation with fractional Brownian motion

Global finite‐dimensional invariant manifolds of the Boissonade system

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

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