I'm looking for a solution of the following discrete-time system:
$x_{t + 1} = A \cdot x_t + B \cdot \exp (x_t) + c$,
where $x_t$ is $n$-dimensional vector, $A$ and $B$ are $n \times n$ constant matrices, and $c$ is $n$-dimensional constant vector. Also, $\exp(\cdot)$ is applied to $x_t$ in entry-wise manner.
I'm interested in finding $f$ such that for any $x_0, A, B, c$ and $t > 0$ it satisfies:
$x_t = f(x_0, A, B, c, t)$
Does the above system can be solved for a closed-form $f$? If so, how can I proceed in finding it?
If not, can it be solved under additional assumptions? For example, if I assume $A$ and $B$ being invertable matrices, would it allow for a closed-form solution? What about the case where $A$ is the identity?
Any suggestions?