The discrete time dynamical system given is $f\left ( x \right )=ax^{2}\left ( x-1 \right )$:
i) Show that $x^{*}=0$ is a stable hyperbolic fixed point.
This is puzzling.
To compute the stability, we look at $\left | f'\left ( x \right ) \right |<1$.
$\left | 2ax-3ax^{2} \right |=0 <1$ so $x^{*}=0$ is stable.
However, the hyperbolicity of a fixed point is determined by the eigenvalue. The eigenvalue is $2ax-3ax^{2}$. At $x^{*}=0$, the eigenvalue is 0 and so the it would be non-hyperbolic in contrast to what is required.
If someone could provide some illumination to my understanding, it would be greatly appreciated.