Nature of Logistic Map for Various Parameter Values

Theorem

Let $f$ denote the logistic map on the closed unit interval $\closedint 0 1$:

$x_{n + 1} = a x_n \paren {1 - x_n}$

where:

$x_0$ can be anywhere in $\closedint 0 1$

$a$ is a real constant in the interval $\hointl 0 4$.

The nature of $f$ depends on the parameter as follows:

Let $0 < a < 1$.

Then the point $x = 0$ is an attracting fixed point for all $x_0 \in \closedint 0 1$.

Let $1 < a < 3$.

Then:

$f$ has a repelling fixed point at $x = 0$ for all $x_0 \in \closedint 0 1$

$f$ has an attracting fixed point at $x = 1 - \dfrac 1 a$ for all $x_0 \in \closedint 0 1$.

Let $3 < a < \sqrt 6$.

Then:

$f$ has a repelling fixed point at $x = 0$ for all $x_0 \in \closedint 0 1$

$f$ has $2$ repelling periodic points of period $2$ for all $x_0 \in \closedint 0 1$.

Let $a > \sqrt 6$.

Then:

$f$ has a repelling fixed point at $x = 0$ for all $x_0 \in \closedint 0 1$

$f$ has repelling periodic points which bifurcate into orbits of periods $2, 4, 8 \ldots$ for all $x_0 \in \closedint 0 1$.