This logistic equation

has a maximum at X_m = 1/2 since g(1/2) = a/4.

g(X) = a x_n (1 -  x_n )  ; g'(X) = a - 2aX and g''(X) = -2a < 0

We have two equilibrium points which are the followings:    X_equil = 0 and X_equil = (a - 1)/a.

g(X) > X  if  0 < X < X_equil  and g(X) < X   if   X_equil  < X < 1

g(X) is strictly monotone increasing function on 0 < X < X_m  and g(X) is strictly monotone
decreasing function on the interval X_m < X < 1.

Also X_m < X_equil  if 2 < a <= 4

So if 1 < a <= 2 from the following theorem called Cull's theorem which says:

    Let A > 0 under some assumptions:

            (i) g(X) is continuous on A with g(0) = 0 and g(X) > 0 if 0 < X < A

            (ii) g(X) > X for 0 < X < X_equil  and g(X) < X for X_equil < X < A

            Then g(X) has a maximum X_m ( 0 < X_m  < X_equil )
            and X_equil is a global attractor if and only if X < g(X) for all X such that X_equil < X < X_m .

The function that helps us to study the global stability of this logistic difference function is called a Lyapunov function and it is equal to :

 V(X) = (X - X_equil )².

In conclusion we have the following graph (where the value is replaced by r):