Abstract
In this paper, stochastic FitzHugh–Nagumo lattice system with nonlinear noise in weighted spaces is considered. Firstly, the well-posedness of solution of such system in a weighted space \(L^2(\Omega ,l^2_\sigma \times l^2_\sigma )\) is established, based on which we further prove the existence and uniqueness of weak pullback mean random attractor in the weighted space. Then the existence and uniqueness of invariant measure are proved in the weighted space \(l^2_\sigma \times l^2_\sigma \) as well as exponentially mixing property in the sense of Wasserstein metric. Moreover, the limit behaviors of invariant measure in the weighted space \(l^2_\sigma \times l^2_\sigma \) are also investigated with respect to noise intensity.
Similar content being viewed by others
References
Chow, S.N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 149(2), 248–291 (1998)
Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572 (1987)
Bates, P.W., Lu, K., Wang, B.: Attractors for lattice dynamical systems. Int. J. Bifurcat. Chaos 11, 143–153 (2001)
Han, X., Kloeden, P.E., Usman, B.: Upper semi-continuous convergence of attractors for a Hopfield-type lattice model. Nonlinearity 33, 1881–1906 (2020)
Caraballo, T., Morillas, F., Valero, J.: Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity. J. Differ. Equ. 253(2), 667–693 (2012)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J . 1, 445–466 (1961)
Bell, J.: Some threshold results for models of myelinated nerves. Math. Biosci. 54, 181–190 (1981)
Jones, C.K.R.T.: Stability of the traveling wave solution of the FitzHugh–Nagumo System. Trans. Am. Math. Soc. 286, 431–469 (1984)
Wang, B.: Dynamical behavior of the almost-periodic discrete FitzHugh–Nagumo systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 1673–1685 (2007)
Boughoufala, A.M., Abdallah, A.Y.: Attractor for FitzHugh–Nagumo lattice equations with almost periodic nonlinear parts. Discret. Contin. Dyn. Syst. Ser. B 26(3), 1549–1563 (2021)
Huang, J.: The random attractor of stochastic FitzHugh–Nagumo equations in an infinite lattice with white noises. Phys. D Nonlinear Phenom. 233, 83–94 (2007)
Gu, A., Li, Y.: Singleton sets random attractor for stochastic FitzHugh–Nagumo lattice equations driven by fractional Brownian motions. Commun. Nonlinear Sci. Numer. Simul. 19, 3929–3937 (2014)
Han, X., Shen, W., Zhou, S.: Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differ. Equ. 250, 1235–1266 (2011)
Bates, P.W., Lu, K., Wang, B.: Attractors of non-autonomous stochastic lattice systems in weighted spaces. Phys. D Nonlinear Phenom. 289, 32–50 (2014)
Li, D., Shi, L., Wang, X.: Long term behavior of stochastic discrete complex Ginzburg–Landau equations with time delays in weighted spaces. Discret. Contin. Dyn. Syst. Ser. B 24(9), 5121–5148 (2019)
Van Vleck, E., Wang, B.: Attractors for lattice FitzHugh–Nagumo systems. Phys. D Nonlinear Phenom. 212, 317–336 (2005)
Wang, X., Zhou, S.: Random attractors for non-autonomous stochastic lattice FitzHugh–Nagumo systems with random coupled coefficients. Taiwan. J. Math. 20(3), 589–616 (2016)
Caraballo, T., Han, X.: Applied Nonautonomous and Random Dynamical Systems: Applied Dynamical Systems. Springer, Cham (2017)
Wang, B.: Weak pullback attractors for mean random dynamical systems in Bochner spaces. J. Dyn. Differ. Equ. 31, 2177–2204 (2019)
Wang, B., Wang, R.: Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise. Stoch. Anal. Appl. 38(2), 213–237 (2020)
Wang, B.: Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise. J. Math. Anal. Appl. 477(1), 104–132 (2019)
Chen, Z., Wang, B.: Weak mean attractors and invariant measures for stochastic Schrödinger delay lattice systems. J. Dyn. Differ. Equ. (2021). https://doi.org/10.1007/s10884-021-10085-3
Wang, R.: Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping. J. Dyn. Differ. Equ. 33, 767–803 (2021)
Wang, B.: Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise. J. Differ. Equ. 268(1), 1–59 (2019)
Wang, R., Guo, B., Wang, B.: Well-posedness and dynamics of fractional FitzHugh–Nagumo systems on \({\mathbb{R}}^N\) driven by nonlinear noise. Sci. China Math. 64, 2395–2436 (2021)
Wang, X., Kloeden, P.E., Han, X.: Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise. Nonlinear Differ. Equ. Appl. 28(43), 1–31 (2021)
Li, D., Wang, B., Wang, X.: Periodic measures of stochastic delay lattice systems. J. Differ. Equ. 272, 74–104 (2021)
Li, D., Wang, B., Wang, X.: Limiting behavior of invariant measures of stochastic delay lattice systems. J. Dyn. Differ. Equ. (2021). https://doi.org/10.1007/s10884-021-10011-7
Chen, Z., Li, X., Wang, B.: Invariant measures of stochastic delay lattice systems. Discret. Contin. Dyn. Syst. Ser. B 26(6), 3235–3269 (2021)
Chen, Z., Wang, B.: Limit measures and ergodicity of fractional stochastic reaction-diffusion equations on unbounded domains. Stoch. Dyn. (2021). https://doi.org/10.1142/S0219493721400128
Eckmann, J.P., Hairer, M.: Invariant measures for stochastic partial differential equations in unbounded domains. Nonlinearity 14, 133–151 (2001)
Kim, J.: Periodic and invariant measures for stochastic wave equations. Electron. J. Differ. Equ. 2004, 1–30 (2004)
Kim, J.: Invariant measures for a stochastic nonlinear Schrödinger equation. Indiana Univ. Math. J. 55, 687–717 (2006)
Kim, J.: On the stochastic Benjamin-Ono equation. J. Differ. Equ. 228, 737–768 (2006)
Brzeźniak, Z., Ondreját, M., Seidler, J.: Invariant measures for stochastic nonlinear beam and wave equations. J. Differ. Equ. 260(5), 4157–4179 (2016)
Misiats, O., Stanzhytskyi, O., Yip, N.K.: Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains. J. Theor. Probab. 29(3), 996–1026 (2016)
Brzeźniak, Z., Motyl, E., Ondrejat, M.: Invariant measure for the stochastic Navier–Stokes equations in unbounded 2D domains. Ann. Probab. 45(5), 3145–3201 (2017)
Chen, Z., Wang, B.: Invariant measures of fractional stochastic delay reaction-diffusion equations on unbounded domains. Nonlinearity 34(6), 3969–4016 (2021)
Chen, L., Dong, Z., Jiang, J., Zhai, J.: On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity. Sci. China Math. 63(8), 1463–1504 (2020)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Hu, H., Xu, L.: Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations. J. Math. Anal. Appl. 466(1), 896–926 (2018)
Acknowledgements
The work is partially supported by the NNSF of China (11471190,11971260), the SDNSF (ZR2014AM002), and the PSF (2012M511488, 2013T60661, 201202023).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chen, Z., Yang, D. & Zhong, S. Limiting Dynamics for Stochastic FitzHugh–Nagumo Lattice Systems in Weighted Spaces. J Dyn Diff Equat 36, 321–352 (2024). https://doi.org/10.1007/s10884-022-10145-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-022-10145-2
Keywords
- Stochastic FitzHugh–Nagumo lattice system
- Weighted space
- Weak pullback mean attractor
- Invariant measure
- Exponential ergodicity