# Iterated function in differential equations

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.

• They are called untractable ;-) – Yves Daoust May 30 '19 at 8:09

$$x(t)=x_0+F^{-1}(t-t_0)$$ where $$F$$ is the antiderivative of $$\dfrac1{f_n}$$.