Discretisation of global attractors for lattice dynamical systems
type: Item
instance of: scholarly article
title: Discretisation of global attractors for lattice dynamical systems
zbMATH Open document ID: 1466.34021
author: Xiaoying Han
author: Peter E. Kloeden
author: Stefanie Sonner
published in: Journal of Dynamics and Differential Equations
publication date: 2020-08-06T00:00:00Z
review text: 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 [\textit{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.
Mathematics Subject Classification ID: 34A33
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 37L60
Mathematics Subject Classification ID: 65L20
zbMATH DE Number: 7229187
zbMATH Keywords: lattice dynamical system
zbMATH Keywords: global attractor
zbMATH Keywords: implicit Euler scheme
zbMATH Keywords: numerical attractor
zbMATH Keywords: upper semicontinuous convergence
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
full work available at URL: s10884-019-09770-1
OpenAlex ID: W2952786453
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Wikidata QID: Q127714483
DOI: 10.1007/S10884-019-09770-1
description: scientific article; zbMATH DE number 7229187