How would one describe $k$ iterations of $\cos(n)$?

Question

What function would one use to describe $k$ iterations of $\cos(n)$? I'm pretty sure that the function would be a damped sine wave (as can be seen in the curve fit equation I wrote in the third row), however the actual formula is probably quite complicated as it involves the Dottie number, which, to my knowledge, cannot be expressed in terms of $e$, $\pi$, or polynomial roots.

Below is an attempt of curve-fitting on $n=1$. Some of the deficiencies of this fit I've noticed while experimenting with Desmos are that the lines are far too steep (even without the scale factor or with a smaller scale factor), and the fit seems to be weaker for even $n$ (although I presume that this is simply an artifact of approximation). Note that the y-axis has been scaled by a factor of 5 for the sake of graph readability.

Answer 1

The limit $\lim_{k\to\infty}\cos^{(k)}(x)$ ($k$ iterations of $\cos$ applied to $x$) is equal to $\xi$, where $\xi$ is a solution to the equation $\cos\xi=\xi$. It is independent of $x$. Therefore for large $k$, you will have lots of oscillations in a small strip around the line $y=\xi\approx 0.739$.

Answer 2

If you look at $\cos(\cos(\cos(x)))$, you’ll see that it’s periodic with period $\pi$. In particular, it looks something like $a \cos(2x)+b$ for some constants $a$ and $b$. In the limit $a$ will geometrically converge to 0 at approximately $\sin(\xi)^k$ while $b$ is $\xi$ such that $\xi=\cos(\xi)$.