Upper semi-continuity of numerical attractors for deterministic and random lattice reversible Selkov systems

In order to inherit numerically the attractor of the (random) reversible Selkov lattice equation, we propose an implicit Euler scheme (IES) in temporal direction to derive the existence of the numerical attractor. We provide a rigorous demonstration of the upper semi-continuous convergence of the numerical attractor to the global attractor when the step size is small enough. Meanwhile, we provide numerical, random, and global attractors with finite-dimensional approximation, thereby obtaining the existence of truncated attractors as the state space dimension tends to infinity. Based on the establishment of existence of random attractor, it is shown that the truncated random attractor converges upper semi-continuously to the truncated global attractor as the intensity of noise goes to zero.