Interesting system of $3$ first-order recursions

In this paper we firstly review how to $\mathit{\text{explicitly}}$ solve a system of $3$ $\mathit{\text{first-order}}\mathit{\text{linear}}\mathit{\text{recursions}}$and outline the main properties of these solutions. Next, via a change of variables, we identify a class of systems of $3$ $\mathit{\text{first-order}}\mathit{\text{nonlinear}}\mathit{\text{recursions}}$ which also are $\mathit{\text{explicitly}}\mathit{\text{solvable}}$. These systems might be of interest for practitioners in $\mathit{\text{applied}}$ sciences: they allow a complete display of their solutions, which may feature interesting behaviors, for instance be $\mathit{\text{completely}}\mathit{\text{periodic}}$ (“isochronous systems”, if the independent variable $n=0,1,2,\mathrm{3...}$is considered a $\mathit{\text{ticking}}\mathit{\text{time}}$), or feature this property $\mathit{\text{only}}\mathit{\text{asymptotically}}$ (as$$$n\to \mathrm{\infty }$).