I'm having trouble understanding why the logistic map becomes unstable when $$\lvert df(x^*)/dx\rvert=1,$$ where $x^*$ are the fixed points of $f(x)=rx(1-x)=x$.
I have read that it can be seen from cobweb diagrams but I don't really understand it.
Any help would be appreciated, thanks.
A fixed point $x^*$ of a differentiable map $f$ is unstable if $|f'(x^*)| > 1$, and asymptotically stable if $|f'(x^*)| < 1$, because if $x_n$ is near $x^*$ and $x_{n+1} = f(x_n)$,
$$ x_{n+1} - x^* = f(x_n) - f(x^*) \approx f'(x^*) (x_n - x^*) $$
EDIT: Suppose $|f'(x^*)| > 1$. If $x_n$ is close to $x^*$ (but not exactly equal), this says $x_{n+1}$ is farther from $x^*$ than $x_n$ is. So in this case the fixed point is unstable: from a starting point close to the fixed point, you go farther and farther away. Conversely, if $|f'(x^*)| < 1$ and your starting point is close to $x^*$, you get closer and closer: the fixed point is asymptotically stable.
If your map depends analytically on a parameter $r$ and you vary $r$, $|f'(x^*)|$ will vary continuously. Thus if you pass from a region where $x^*$ is stable to one where it is unstable (or vice versa), $|f'(x^*)|=1$ at the transition point.
It is possible in principle to have $|f'(x^*)| = 1$ at a certain $r$ and $<1$ on both sides of $r$, or $> 1$ on both sides of $r$, so $|f'(x^*)|=1$ doesn't necessarily signal a transition between stable and unstable.