Logistic Map

You can move the slider for different values of

$\mu$ . The intersection of the red line with the background image shows all possible points in the long run.

Stability of the two fixed point

$x^*=0$ and

$x^*=1-1/\mu$ with the Jacobian

$f_\mu'(x^*) = \mu(1-2x^*)$ :

For the fixed point $x^*=0, f_\mu'(x^*) = \mu \Longrightarrow$ stable only for $\mu < 1$
For the fixed point $x^*=1-1/\mu, f_\mu'(x^*) = 2-\mu \Longrightarrow$ stable only for $|2-\mu|<1$ or $1<\mu<3$ .

$\mu$ passes

$3$ ,

$f_\mu'(x^*)|_{x^*=1-1/\mu}$ passes

$-1$ , indicating period doubling bifurcation.

Different behaviours of the logistic map xn+1=fμ(xn)=μxn(1−xn) x_{n+1} = f_\mu(x_n) = \mu x_n(1-x_n)