I am currently working on a problem that involves differential equations that contains iterated functions. The problem can be described as that one seeks the solution for the equation
$$ \dot{x} = f^{n}(x) $$
where the right-hand side is an iterated function. A simple example can be the equation
$$ \dot{x} = \textrm{sin}(\textrm{sin}(x)) $$ with $f = \textrm{sin}(x)$ and $n = 2$.
My issue is that I could not find any materials on these types of equations, because I am not even sure what they are called. As such the question is if there is any proper material on this topic that might contain solution methods or some general theorems. I am using these iterated functions in adaptive control and they seem to be very effective in treating parametric disturbances of nonlinear systems which makes them interesting to analyze.
The solution of your equation, which is separable, is "simply"
$$x(t)=x_0+F^{-1}(t-t_0)$$ where $F$ is the antiderivative of $\dfrac1{f_n}$.
I don't think that you will find much theory specific to this particular kind of equation (unless they are frequent in your application domain). I would have a look at the properties of iterated functions in general.