This may be somewhat of an underspecified question but I'll nonetheless give it a try. In the context of applied work, I've recently come across systems of the form
$$ \mathbf{x}_{t+1}=\mathbf{A}\mathbf{x}_t,$$
where $\mathbf{x}_{t}\in\mathbb{R}^n$ and $\mathbf{A}\in\mathbb{R}^{n\times n}$ (more specifically, $\mathbf{A}$ is row-stochastic). Of course, this system is not very interesting and can be rewritten as
$$ \mathbf{x}_t = \mathbf{A}^t\mathbf{x}_0$$
and then asymptotics are determined by the behavior of $\lim_{t\rightarrow\infty} \mathbf{A}^t$. If instead of this process, I would be interested in a process
$$ \mathbf{x}_{t+1}=f_{\mathbf{A},\mathbf{Q}}(\mathbf{x}_t),$$
where the operator $f$ has a form that is `close' to linear, namely,
$$\Bigl(f_{\mathbf{A},\mathbf{Q}}(\mathbf{x})\Bigr)_i=\sum_{j=1}^n A_{ij}Q_{ij}(x_j),$$
whereby $\mathbf{Q}$ is a matrix of arbitrary functions (if all functions in $\mathbf{Q}$ are identity functions, the original system is retrieved) - i.e. $Q_{ij}:\mathbb{R}\rightarrow\mathbb{R}$ - what kind of literature would I have to consult? Has anyone ever seen such a system as above?
I know that this question is pretty vague, but any help would be appreciated.