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 \( f_\mu'(x^*) = \mu(1-2x^*)\):*period doubling* bifurcation.

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 \(1<\mu<3\).