$U_{n+1} = a * U_n * (1-U_n)$ with : $3<a<3.6$ What are some values of $a$ such that $U_n$ changes its periodicity?

Question

$$U_{n+1} = a \cdot U_n \cdot (1-U_n) = a \cdot U_n - a \cdot (U_n)^2$$

with : $$3<a<3.6$$

What are some values of $a$ such that $U_n$ changes its periodicity?

I've computed $U_n$ for many different values of $a$, but $U_n$ always has a period of $2$.

For example here with $a=3.3$ :

Answer 1

Find a book that has this diagram:

In fact, $3.3$ is in the period $2$ region.

Here is a table, period-doublings at:

$3, 3.4494897, 3.5440903, 3.5644073, 3.5687594, 3.5696916, \dots$