Edited the post and question after orion's comment; (My first encounter with a chaotic system)
Seems that the Natural Logarithm brings chaos as a recursive function:
$$ f(n+1)=\ln{|f(n)|}$$$$f(0)=|k|, k\in\mathbb R\setminus{\{e,1\}}$$
How can we determine how far are the stable points of some lengths away from some $f_k(n)?$
Example graph
If we plot the graph for $k=2$ for example, we have the following images:
Second graph being the joined points of the values plotted in the first graph.
We can change $k$ by an arbitrary small number, and that will affect the entire plot almost completely, meaning its very sensitive which will recursively produce chaos.
Stability
The points jump around $y=-W(1)$.
If we set $k=W(1)$, then $f(n)=-W(1)$ for all $n>0$; the entire graph is a stable line.
We can have a sequence of negative values of any arbitrary length, by picking $k=W(1)-\omega$, where $\omega$ is some real number. The smaller the $\omega$, the more negative values will appear before the first positive one. For example:
$\omega=10^{-20}$, gives a stable line of $81$ negatives values before the first positive peak.
If $\omega=10^{-t}$, then I can estimate the number of negative values before the first positive:
$$ N=4t+2\left\lfloor\frac{t+19}{33}\right\rfloor-1 $$
This seems to hold exactly for almost all values of $t$. To put this into perspective; For $t\le200$, it's only off by $1$ for only few values, which are $t=13,46,179$. I'm not sure why this estimation seems to work. The $t$ is a positive integer.
Can the function reach a stable period of $N$ consecutive negative values, for every initial value of $k$, if $N$ is some arbitrary positive integer? Otherwise, could one determine which lengths are reachable by some $k$?
How could one show something like this?
I suppose this is equivalent to finding the set of values which is reachable by this recursive function, in condition to initial $k$.
Longest period for $k=2$ I've found so far, is $16$, appearing at $n=289$;
Is it possible to predict where the stable periods of some lengths $N$ will appear?
The closer some $f_k(n)$ is to $W(1)$, the longer stable period will be produced afterwards.
How can we define a measure of how close the $f_k(n)\approx W(1)$ approximation is? And then use that measure to estimate $N$ at that $f_k(n)$?
This question boils down to;
How would one find a way to estimate how many iterations is necessary to reach some length $N$ of consecutive negative values, from some $f_k(n)$?
As an enthusiastic layman of maths who is finishing high school at the moment, I'm not sure how hard or easy this question even is, as this is my first time ever being interested in it, and without any specific or related knowledge.