Consider the dynamical system \begin{align} x[k+1]=f [x[k], u[k]] \end{align} where $x[k]\in \mathbb R^n, u: \mathbb N\bigcup\{0\}\to \mathbb R^p$
and
\begin{align} x[k+1]=f [x[k], u[x[k]]] \end{align} where $x[k]\in \mathbb R^n, u: \mathbb R^n\to [0,\infty)$
Could anyone explain to me what is the difference between these two systems? Can I still think that the second system is with input, but the input is a function of the state variables? Can anyone give me some examples of the system described in the end?