IDEA: Let's start with the Logistic differential equation given by
where y=y(t). By definition of the derivative as a difference quotient, we have that, for small h > 0,
from which we derive that
Now we fix the, so called, step-size, h, with the value h=1 and set t=n a natural number. Writing 
for each n, we find
that (3) gives
or
where we take it that 
. At this point we make a change of dependent variable by requiring that
for every n. A glance at the new equation for 
shows us that
which, when interpreted as an equality gives us the iterations of the logistic map
Generally, 
and 