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.

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    $\begingroup$ They are called untractable ;-) $\endgroup$ – Yves Daoust May 30 '19 at 8:09

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.


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