Does a closed loop system with a reference of its previous iteration always reach a point in which it adventually loops, even if that loop is stable? For example you have a $F(\cdot)$ which starts off taking some input $a$, which describes some set of objects and their states, and creating a output $F(a)$, which can be renamed $F_1$, that also describes a set of objects and their states, which is then input into $F$, to get $F_2 = F(F_1)$, or $F(F(a))$. 

If this is repeated for an extensive period is it guaranteed that that the values of the individual iterations will be loop? For example $F(F_{x+n})=F(F_{x+2n})$ etc and $F(F_{x+m})=F(F_{x+2m})$ where $x$ is the start of the cycle and $m$ and $n$ are the frequency of the cycle.
As is usual with these sort of questions, the answer is "it depends".

For example, suppose
$$
F(x) = x
$$
then the loop is immediate.

For example,  suppose
$$
F(x) = x/2
$$
then the loop never occurs unless $x = 0$.

The same will be true for more complex $F$s. 

For example
$$
F(x) = \mathbf{A} x
$$
where $\mathbf{A}$ is a matrix. If $\mathbf{A}$ is a rotation matrix of a certain type (so that it rotates by $360/n$ degrees where $n$ is an integer) then the period will be $n$. If $\mathbf{A}$ is something else, then all bets are off.