Recently I've been attempting to write and explicit form of the logistic map:
$x_{n+1} = rx_n(1-x_n)$
where $x$ is between $0$ and $1$
It's a pretty simple equation, but I was curious if it was possible to form an explicit equation to solve for $x_n$, that way it could be re-worked to create a functional relationship between $r$ and the stabilization value the function arrives at, let's call $s$. This function is famously part of chaos theory and creates bifurcation when comparing $r$ and $s$. At a growth rate of $3$, $s$ branches into having two values it oscillates between. As you go further, the bifurcation keeps doubling the number of answers until it becomes totally chaotic around an $r$ value of $4$. The stabilization value I would assume could be determined using an infinite sum, but I'd need to get the equation in an explicit form before re-arranging it.
I'm familiar with ways to basically convert recursive and explicit equations, but this one has me stumped. I attempted writing out the solutions with different starting values for $x$ and couldn't find an obvious pattern in how the values changed. I should note that I feel as though I should assume the final result would have some constant added to it to represent the initial value, but I'm not sure on that. I also attempted to write out equations to solve for specific values of $n$, but the number of terms involved increased rapidly and as well I personally couldn't recognize a pattern.
The most recent paper I can find on this is from 1998 and that seems a shame. As well to be honest it is way over my head, but it may help:
"An explicit solution for the logistic map"
I have some theories as to the importance of the equations, but they're really just hunches, and probably misguided. Either way I'm curious if this type of problem is even solvable, and I appreciate any help you guys can give.
Thanks!