Logistic map confusion Logistic map is a simple example of discrtete dynamical systems defined as $$x_{i+1}=\lambda*x_{i}*(1-x_{i}).$$
It is known that for $\lambda=4$ this map shows a chaotic behavior for $x\in (0,1)$. 
The question:
How could it be that the logistic map at $\lambda=4$ is chaotic, while,  starting with $x_0=0.5$, the iteration goes to 1 then fixed at 0?

What is actually meant with "chaotic" is that almost all start values lead to a (theoretically) non-periodical sequence and the sequence shows no structure which means that it can be used as a pseudo-random-generator.
