ATTRACTORS FOR RANDOM LATTICE DYNAMICAL SYSTEMS WITH INFINITE MULTIPLICATIVE COLORED NOISE

Meng Gao; Anhui Gu

doi:10.11948/20220343

2023 Volume 13 Issue 5

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Meng Gao, Anhui Gu. ATTRACTORS FOR RANDOM LATTICE DYNAMICAL SYSTEMS WITH INFINITE MULTIPLICATIVE COLORED NOISE[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2433-2451. doi: 10.11948/20220343

Citation:

Meng Gao, Anhui Gu. ATTRACTORS FOR RANDOM LATTICE DYNAMICAL SYSTEMS WITH INFINITE MULTIPLICATIVE COLORED NOISE[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2433-2451. doi: 10.11948/20220343

ATTRACTORS FOR RANDOM LATTICE DYNAMICAL SYSTEMS WITH INFINITE MULTIPLICATIVE COLORED NOISE

Meng Gao^1,,
Anhui Gu^1, ,

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Author Bio: Email: gmeng1@email.swu.edu.cn(M. Gao)
Corresponding author: Email: gahui@swu.edu.cn(A. Gu)

Abstract

Abstract

In this paper, we establish the existence and uniqueness of random attractor for the first-order random lattice differential equation with a nonlinear colored noise at each node. We first rewrite the equation as a random evolution system and then prove the existence of a unique weak solution. Finally, we obtain the existence of a unique random attractor for the underlying random dynamical system.
- Random lattice differential equation /
- random dynamical system /
- random attractor /
- colored noise
MSC: 60H15, 37L60, 35B41

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References

[1]	C. Aliprantis and K. Border, Infinite Dimensional Analysis: A Hitchhikers Guide, Springer, Berlin, 2007. Google Scholar
[2]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. Google Scholar
[3]	P. Bates, B. Wang and K. Lu, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2001, 11, 143–153. doi: 10.1142/S0218127401002031 CrossRef Google Scholar
[4]	P. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 2006, 6, 1–21. doi: 10.1142/S0219493706001621 CrossRef Google Scholar
[5]	H. Bessaih, M. Garrido-Atienza, X. Han and B. Schmalfuss, Stochastic lattice dynamical systems with fractional noise, SIAM J. Math. Anal., 2017, 49, 1495–1518. doi: 10.1137/16M1085504 CrossRef Google Scholar
[6]	T. Caraballo, M. Garrido-Atienza, B. Schmalfuss and J. Valero, Non–autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 2008, 21, 415–443. doi: 10.3934/dcds.2008.21.415 CrossRef Google Scholar
[7]	T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 2016, 130, 255–278. doi: 10.1016/j.na.2015.09.025 CrossRef Google Scholar
[8]	T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 2008, 3, 317–335. doi: 10.1007/s11464-008-0028-7 CrossRef Google Scholar
[9]	S. N. Chow, Lattice Dynamical Systems, Lecture Notes in Math., 1822, Springer, Berlin, 2003, 1–102. Google Scholar
[10]	S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 1995, 42, 746–751. doi: 10.1109/81.473583 CrossRef Google Scholar
[11]	L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 1993, 40, 147–156. Google Scholar
[12]	F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 1996, 59, 21–45. doi: 10.1080/17442509608834083 CrossRef Google Scholar
[13]	A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. B, 2019, 24, 5737–5767. Google Scholar
[14]	A. Gu and B. Wang, Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 2018, 23, 1689–1720. Google Scholar
[15]	M. Hilbert, A solid-solution model for inhomogeneous systems, Acta Metall., 1961, 9, 525–535. doi: 10.1016/0001-6160(61)90155-9 CrossRef Google Scholar
[16]	O. Kallenberg, Foundations of Modern Probability, Springer-Verlag, New York, 1997. Google Scholar
[17]	R. Kapral, Discrete models for chemically reacting syetems, J. Math. Chem., 1991, 6, 113–163. doi: 10.1007/BF01192578 CrossRef Google Scholar
[18]	J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 1987, 47, 556–572. doi: 10.1137/0147038 CrossRef Google Scholar
[19]	L. Ridolfi, P. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, New York, 2011. Google Scholar
[20]	G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. Google Scholar
[21]	G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 1930, 36, 823–841. doi: 10.1103/PhysRev.36.823 CrossRef Google Scholar
[22]	M. Wang and G. Uhlenbeck, On the theory of Brownian motion. II, Rev. Modern Phys., 1945, 17, 323–342. doi: 10.1103/RevModPhys.17.323 CrossRef Google Scholar
[23]	X. Wang, J. Shen, K. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 2021, 280, 477–516. doi: 10.1016/j.jde.2021.01.026 CrossRef Google Scholar
[24]	M. Zgurovsky, P. Kasyanov, O. Kapustyan, J. Valero and N. Zadoianchuk, Attractors for Lattice Dynamical Systems, In: Evolution Inclusions and Variation Inequalities for Earth Data Processing Ⅲ. Advances in Mechanics and Mathematics, Springer, Berlin, 2012, 27. Google Scholar