Is there a specific name for a dynamical system that depends on the relative indexation $i\pm k$ for some $k$? For example, consider the following dynamical system defined on a ring of cells by $$ \begin{align} \dot{u}_i&=u_i+2(v_{i-1}+v_{i+1})\\ \dot{v}_i&=u_i-v_i \end{align} $$ for each cell $i$, where the derivative is with respect to time $t$.
The main reason I ask this is because I won't to compare this kind of systems with systems involving spatial coordinates, $u(x,t)$, as in reaction-diffusion equations.
The ODE system is translationally invariant, if $i \in \mathbb{Z}$ or the indices have periodic boundary conditions (identifying $i=N+1$ with $i=1$), meaning that no unit $i$ is distinguished. Translation invariance implies that the system specified in matrix form contains a (block) circulant matrix. In addition, the circulant is banded, as stated by Ian.