Proving non-repitition of a sequence I have heard that the sequence $$x_{n+1}=rx_n(1-x_n)$$ for $r$ between $3$ and $4$ does not recur i.e. there is no $a>0$ such that $x_{n+a} = x_n$ and $x_0$ is any number between 0 and 1 exclusive. I have tried it by brute force up to $x_{120}$ and it does not seem to recur, but is it possible that the 'apparent' non-repetition was due to round-off errors and that it actually recurs? How would I be able to prove its recurrence/non-recurrence?

That is simply not true.

Indeed for $r=4$, the graph of $f^n=\underbrace{f\circ f\circ \cdots\circ f}_{n\text{ times}}$ is composed of $2^{n-1}$ arches with $y$-coordinates $0\rightarrow 1\rightarrow 0$.

Therefore the graph of $y=f^n$ and $y=x$ has $2^n$ intersections and so $2^n$ period-$n$ points.

In fact the periodic points of that map are dense in $[0,1]$ for $r=4$.