In this work, we examine first-order lattice dynamical systems, which are discretized versions of reaction–diffusion equations on the real line. We prove the existence of a global attractor in \(\ell ^2\), and using the method by Chueshov and Lasiecka (Dynamics of quasi-stable dissipative systems, Springer, Berlin, 2015; Memoirs of the American Mathematical Society, vol 195(912), AMS, 2008), we estimate its fractal dimension. We also show that the global attractor is contained in a finite-dimensional exponential attractor. The approach relies on the interplay between the discretized diffusion and reaction, which has not been exploited as yet for the lattice systems. Of separate interest is a characterization of positive definiteness of the discretized Schrödinger operator, which refers to the well-known Arendt and Batty’s result (Differ Int Equ 6:1009–1024, 1993).

中文翻译：

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晶格动力学系统：整体和指数吸引子的耗散机理和分形维数

在这项工作中，我们研究了一阶晶格动力学系统，它是实线上反应扩散方程的离散版本。我们证明\（\ ell ^ 2 \）中存在全局吸引子，并使用Chueshov和Lasiecka的方法（准稳态耗散系统动力学，Springer，柏林，2015年；美国数学学会回忆录，第195（912）卷，AMS，2008年），我们估计了它的分形维数。我们还表明，全局吸引子包含在有限维指数吸引子中。该方法依赖于离散的扩散和反应之间的相互作用，而晶格系统尚未利用该相互作用。值得关注的是离散Schrödinger算子的正定性的表征，它是指著名的Arendt和Batty的结果（Differ Int Equ 6：1009–1024，1993）。