×

Discretisation of global attractors for lattice dynamical systems. (English) Zbl 1466.34021

The authors investigate the existence of numerical attractors for lattice dynamical system (LDS) \[\frac{du_i}{dt}=\nu(u_{i-1}-2u_i+u_{i+1})+f(u_i)+g_i,\ \ \ i\in\mathbb{Z},\ \ \nu>0,\tag{1}\] where \((g_i)_{i\in\mathbb{Z}}\in{\ell}^2\), \({\ell}^2=\big\{u=(u_i)_{i\in\mathbb{Z}}:\sum_{i\in\mathbb{Z}}u_i^2<\infty\big\}\) is a Hilbert space, the nonlinearity \(f\colon\mathbb{R}\to\mathbb{R}\) is a continuously differentiable function satisfying inequality \(f(s)s\le-\alpha s^2\) for some \(\alpha>0\).
The LDS (1) can be written as the infinite dimensional ordinary differential equation in \({\ell}^2\) \[\frac{du}{dt}=Au+F(u)+g,\tag{2}\] where \(A\colon{\ell}^2\to{\ell}^2\), \((Au)_i=\nu(u_{i-1}-2u_i+u_{i+1})\), \(F\colon{\ell}^2\to{\ell}^2\) is the Nemytskii operator, \(F(u)=(f(u_i))_{i\in\mathbb{Z}}\), \(g=(g_i)_{i\in\mathbb{Z}}\).
For given initial data \(u(0) = u_0\in{\ell}^2\) the existence and uniqueness of a global solution of the ODE system (2) follows by standard argument (see [P. W. Bates et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, No. 1, 143–153 (2001; Zbl 1091.37515)]). The semi-dynamical system generated by (2) possesses a global attractor.
By the implicit Euler scheme with constant time step \(h>0\), the ODE (2) is reduced to \[u^{(h)}_{n+1}=u^{(h)}_{n}+hAu^{(h)}_{n+1} +hF(u^{(h)}_{n+1})+hg, \ \ n\in\mathbb{N}_0, \tag{3}\] where \(u^{(h)}_{n}=\big(u^{(h)}_{n,i}\big)_{i\in\mathbb{Z}}\).
The finite dimensional truncations of (3) are considered. It is proved that the finite dimensional numerical attractors converge upper semicontinuously to the global attractor of the original lattice model (1) as the discretisation step size tends to zero.

MSC:

34A33 Ordinary lattice differential equations
34D45 Attractors of solutions to ordinary differential equations
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
65L20 Stability and convergence of numerical methods for ordinary differential equations

Citations:

Zbl 1091.37515
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abdallah, AY, Uniform global attractors for first order non-autonomous lattice dynamical systems, Proc. Am. Math. Soc., 138, 3219-3228 (2010) · Zbl 1204.37077
[2] Bates, PW; Lu, K.; Wang, B., Attractors for lattice dynamical systems, Int. J. Bifur. Chaos, 11, 143-153 (2001) · Zbl 1091.37515
[3] Chow, S-N; Mallet-Paret, J., Pattern formation and spatial chaos in lattice dynamical systems: I, IEEE Trans. Circuits Syst., 42, 746-751 (1995)
[4] Chua, LO; Yang, L., Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35, 1257-1272 (1988) · Zbl 0663.94022
[5] Grüne, L., Asymptotic Behaviour of Dynamical and Control Systems Under Perturbation and Discretization, Springer LNM 1783 (2002), Heidelberg: Springer, Heidelberg
[6] Han, X., Asymptotic dynamics of stochastic lattice differential equations: a review, Continuous and distributed systems II, Stud. Syst. Decis. Control, 30, 121-136 (2015) · Zbl 1335.37033
[7] Han, X.; Kloeden, PE, Attractors under Discretisation (2017), Cham: Springer, Cham
[8] Jentzen, A.; Kloeden, PE, Taylor Approximations of Stochastic Partial Differential Equations, CBMS Lecture series (2011), Philadelphia: SIAM, Philadelphia · Zbl 1240.35001
[9] Kapral, R., Discrete models for chemically reacting systems, J. Math. Chem., 6, 113-163 (1991)
[10] Keener, JP, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47, 556-572 (1987) · Zbl 0649.34019
[11] Kloeden, PE; Lorenz, J., Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Analysis, 23, 986-995 (1986) · Zbl 0613.65083
[12] Stuart, AM; Humphries, AR, Numerical Analysis and Dynamical Systems (1996), Cambridge: Cambridge University Press, Cambridge
[13] Wang, B., Dynamics of systems on infinite lattices, J. Differ. Equ., 221, 224-245 (2006) · Zbl 1085.37056
[14] Zhou, S., Attractors and approximations for lattice dynamical systems, J. Differ. Equ., 200, 342-368 (2004) · Zbl 1173.37331
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.