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?
 A: Chaotic maps may still have fixed points and periodic orbits. What distinguishes a chaotic map is that points that start out close together quickly diverge from one another - this is called "sensitive dependence on initial conditions".
For the logistic map $x_{i+1}=4x_i(1-x_i)$ the sequence $\{x_i\}$for an initial value $x_0=0.5$ is, as you say,
$\{0.5, 1, 0, 0, \dots \}$
but for $x_0=0.51$ we get the sequence
$\{0.51, 0.996, 0.00159936, 0.00637208, 0.025385647, \dots \}$
and for $x_0=0.52$ we get the sequence
$\{0.52, 0.9984, 0.00638976, 0.025395724, 0.099003124 \dots \}$
and by the time we get to $x_{10}$ the two sequences are completely different:

Contrast this with the behaviour when $\lambda=2$, when $x_{i+1}=2x_i(1-x_i)$:
$\{0.5, 0.5, 0.5, 0.5, \dots \} \\
\{0.51, 0.4998, 0.49999992, 0.5, \dots \} \\
\{0.52, 0.4992, 0.49999872, 0.5, \dots \}$
when initial points that start close to $0.5$ all rapidly converge to the fixed point $0.5$.
A: 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.
