September  2022, 27(9): 5205-5224. doi: 10.3934/dcdsb.2021271

Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

*Corresponding author: Xiaohu Wang

Received  May 2021 Revised  September 2021 Published  September 2022 Early access  November 2021

Fund Project: This work was supported by NSFC (11871049 and 12090013) and Young crop project of Sichuan University (2020SCUNL111)

In this paper, we study the asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. The considered systems are driven by the fractional discrete Laplacian, which features the infinite-range interactions. We first prove the existence of pullback random attractor in $ \ell^2 $ for stochastic lattice systems. The upper semicontinuity of random attractors is also established when the intensity of noise approaches zero.

Citation: Yiju Chen, Xiaohu Wang. Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5205-5224. doi: 10.3934/dcdsb.2021271
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. C. Bressloff, Waves in Neural Media: From Single Neurons to Neural Fields, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, New York, 2014. doi: 10.1007/978-1-4614-8866-8.

[3]

P. W. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[4]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[5]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[6]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

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L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits and Systems, 40 (1993), 147-156.  doi: 10.1109/81.222795.

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L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans. Circuits and Systems, 35 (1998), 1257-1272.  doi: 10.1109/31.7600.

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Ó. CiaurriT. A. GillespieL. RoncalJ. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6.

[11]

Ó. CiaurriC. LizamaL. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Lett., 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007.

[12]

Ó. Ciaurri and L. Roncal, Hardy's inequality for the fractional powers of a discrete Laplacian, J. Anal., 26 (2018), 211-225.  doi: 10.1007/s41478-018-0141-2.

[13]

Ó. CiaurriL. RoncalP. R. StingaJ. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 330 (2018), 688-738.  doi: 10.1016/j.aim.2018.03.023.

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S. N. Chow, Lattice dynamical systems, Dynamical Systems, Lecture Notes in Math., 1822 (2003), 1-102.  doi: 10.1007/978-3-540-45204-1_1.

[15]

S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751.  doi: 10.1109/81.473583.

[16]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[17]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[18]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[19]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[20]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.

[21]

C. GuoY. ChenJ. Shu and X. Yang, Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains, Front. Math. China, 16 (2021), 59-93.  doi: 10.1007/s11464-021-0896-7.

[22]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[23]

X. HanP. E. Kloeden and B. Usman, Long term behavior of a random Hopfield neural lattice model, Commun. Pure Appl. Anal., 18 (2019), 809-824.  doi: 10.3934/cpaa.2019039.

[24]

X. HanP. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.

[25]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[26]

Y. Hong and C. Yang, Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit, SIAM J. Math. Anal., 51 (2019), 1297-1320.  doi: 10.1137/18M120703X.

[27]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317 (2013), 563-591.  doi: 10.1007/s00220-012-1621-x.

[28]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[29]

C. Lizama and L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 1365-1403.  doi: 10.3934/dcds.2018056.

[30]

D. LiB. Wang and X. Wang, Random dynamics of fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$ without uniqueness, J. Math. Phys., 60 (2019), 072704.  doi: 10.1063/1.5063840.

[31]

D. LiX. Wang and J. Zhao, Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process, Commun. Pure Appl. Anal., 19 (2020), 2751-2776.  doi: 10.3934/cpaa.2020120.

[32]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.

[33]

C. Martínez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Studies 187, Amsterdam, 2001.

[34]

M. SuiY. WangX. Han and P. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008.

[35]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[36]

W. M. Schouten and H. J. Hupkes, Nonlinear stability of pulse solutions for the discrete Fitzhugh-Nagumo equation with infinite-range interactions, Discrete Contin. Dyn. Syst., 39 (2019), 5017-5083.  doi: 10.3934/dcds.2019205.

[37]

B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006.

[38]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[39]

X. WangP. Kloeden and X. Han, Attractors of Hopfield-type lattice models with increasing neuronal input, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 799-813.  doi: 10.3934/dcdsb.2019268.

[40]

X. WangJ. ShenK. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026.

[41]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[42]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.

[43]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55.  doi: 10.1016/j.jmaa.2012.04.080.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. C. Bressloff, Waves in Neural Media: From Single Neurons to Neural Fields, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, New York, 2014. doi: 10.1007/978-1-4614-8866-8.

[3]

P. W. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[4]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[5]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[6]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[8]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits and Systems, 40 (1993), 147-156.  doi: 10.1109/81.222795.

[9]

L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans. Circuits and Systems, 35 (1998), 1257-1272.  doi: 10.1109/31.7600.

[10]

Ó. CiaurriT. A. GillespieL. RoncalJ. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6.

[11]

Ó. CiaurriC. LizamaL. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Lett., 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007.

[12]

Ó. Ciaurri and L. Roncal, Hardy's inequality for the fractional powers of a discrete Laplacian, J. Anal., 26 (2018), 211-225.  doi: 10.1007/s41478-018-0141-2.

[13]

Ó. CiaurriL. RoncalP. R. StingaJ. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 330 (2018), 688-738.  doi: 10.1016/j.aim.2018.03.023.

[14]

S. N. Chow, Lattice dynamical systems, Dynamical Systems, Lecture Notes in Math., 1822 (2003), 1-102.  doi: 10.1007/978-3-540-45204-1_1.

[15]

S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751.  doi: 10.1109/81.473583.

[16]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[17]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[18]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[19]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[20]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.

[21]

C. GuoY. ChenJ. Shu and X. Yang, Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains, Front. Math. China, 16 (2021), 59-93.  doi: 10.1007/s11464-021-0896-7.

[22]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[23]

X. HanP. E. Kloeden and B. Usman, Long term behavior of a random Hopfield neural lattice model, Commun. Pure Appl. Anal., 18 (2019), 809-824.  doi: 10.3934/cpaa.2019039.

[24]

X. HanP. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.

[25]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[26]

Y. Hong and C. Yang, Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit, SIAM J. Math. Anal., 51 (2019), 1297-1320.  doi: 10.1137/18M120703X.

[27]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317 (2013), 563-591.  doi: 10.1007/s00220-012-1621-x.

[28]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[29]

C. Lizama and L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 1365-1403.  doi: 10.3934/dcds.2018056.

[30]

D. LiB. Wang and X. Wang, Random dynamics of fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$ without uniqueness, J. Math. Phys., 60 (2019), 072704.  doi: 10.1063/1.5063840.

[31]

D. LiX. Wang and J. Zhao, Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process, Commun. Pure Appl. Anal., 19 (2020), 2751-2776.  doi: 10.3934/cpaa.2020120.

[32]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.

[33]

C. Martínez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Studies 187, Amsterdam, 2001.

[34]

M. SuiY. WangX. Han and P. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008.

[35]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[36]

W. M. Schouten and H. J. Hupkes, Nonlinear stability of pulse solutions for the discrete Fitzhugh-Nagumo equation with infinite-range interactions, Discrete Contin. Dyn. Syst., 39 (2019), 5017-5083.  doi: 10.3934/dcds.2019205.

[37]

B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006.

[38]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[39]

X. WangP. Kloeden and X. Han, Attractors of Hopfield-type lattice models with increasing neuronal input, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 799-813.  doi: 10.3934/dcdsb.2019268.

[40]

X. WangJ. ShenK. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026.

[41]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[42]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.

[43]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55.  doi: 10.1016/j.jmaa.2012.04.080.

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