I have a question about the period orbits of the logistic map $f(x)=r \cdot x(1-x), r \in [0,4], x \in [0,1]$. The bifurcation-diagram own for $r<3.5699$ only periodic orbits of period $ p=2^k,k \in \mathbb{N}$. These period-doubling is a result of pitchfork-bifurcations. So far so clear. My question is about the periodic orbits of odd periods e.g. $5,7,9,11,...$
The periodic orbit of period $3$ arises through a saddle-node bifurcation. That is, when a pair of fixed points collapse and leave behind a single fixed point of higher period. In the paper "The logistic map and the birth of Period-3 Cycle" by Alberto, Ariel, and Yu.A. $(2012)$ one can see the birth of the periodic orbit of period $3$ through a tangent bifurcation at $r = 1 + 2 \sqrt2$.
So far so good. But I have tho following questions:
$1$: Why can't we see in the following bifurcation-diagram of the logistic map 
$3$ orbit." /> that for $r$ just a little bit smaller than $1 + 2 \sqrt2$ that we have a declining number of periodic points but the birth of the period $3$ orbit is abruptly? Because for $r<1+2 \sqrt2$ we have many periodic points and I thought that a saddle-node bifurcation only reduces the number of fixed points by half. So how is it possible, that all of the sudden we have an orbit of period $3$?
$2$: Are the stable periodic orbits of period $5,7,9,11,...$ formed analogous to the orbit of period $3$? And is there any result that proofs the existence of periodic orbits for any odd number?
I refer you to the theorem by Sharkovskii, which gives a certain order relation for periodic orbits. This is related to the famous result by Li and Yorke, known as 'period 3 implies chaos'.
Basically - once you have period $3$, you have periods of all lengths. More generally, Sharkovskii's theorem gives you a characterization of which periodic orbits you may have, based on the existence of a "maximal" orbit in the order relation. The orbits located lowest in the order relation are the powers of two, which can be seen from the bifurcation diagram. The periodic orbits of odd order are highest in the order relation, and if you have a periodic orbit of odd order, then you will have orbits of all even orders.