Logistic map plot is incorrect (Matlab)

I am trying to iterate and plot the Logistic map nonlinear function given by the equation: x[n+1] = 4*x[n]*(1-x[n]). I have found an implementation of the function here https://www.mathworks.com/matlabcentral/answers/122101-plotting-f-x-as-a-function-of-x-logistic-map

I have followed the same thing only the number of points N and the starting initial condition is different in my implementation. I don't know why I am getting no values in the output; mostly are zero value. When the initial condition is x[1] = 0.5, I get the weird plot as given in the picture. But when the initial condition is say 0.3, then I get the proper Logistic map. Theoretically, the initial condition can be any number between 0 and 1. So, why does the code not work when the initial condition is 0.5?

What is the problem?

N=20000; % number of data points
x = zeros(1,N);
x(1) = 0.5; % initial condition (can be anything from 0 to 1)


for n = 1:N
    x(n+1) = 4*x(n)*(1-x(n));
end
 plot(x(1:N),x(2:N+1),'rs-')
 xlabel('x_n')
 ylabel('x_{n+1}')

Here is the plot

Solution

The result you're getting is correct. Let me explain it with the cobweb diagram below (source: http://sites.saintmarys.edu/%7Esbroad/example-logistic-cobweb.html)

The parabola is the curve y = 4*x*(1-x), the blue linear curve is y=x. The way the points x[n] are determined is the following:

start at point (x0,y(x0)) (often the line with (x0,0) is drawn)
go horizontal until you hit the linear curve
go vertical until you hit the parabola. This is your x[n+1].
repeat steps 2-3 until infinity or x[n]=x[n-1] (stable point) or x[n]=x[n-m] (periodicity)

Note the stable points are where the parabola and linear curve intersect.

Applying this method to with your parameters gives:

Start in (0.5,1)
go horizontal to (1,1)
go vertical to (1,0) (this is x[1])
go horizontal to (0,0)
go vertical to (0,0) (this is x[2])
go horizontal to (0,0)
go vertical to (0,0) (this is x[3])
stop

So you just happen to end up the stable point.