Interesting system of $3$ first-order recursions

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