= Solution
Let $f(u)=au(1-u^2)$. A <fixed point> satisfies $u=f(u)$, so
$$
u=0\quad\hbox{or}\quad u_\pm=\pm\sqrt{1-\frac1a}.
$$
The nonzero fixed points exist for $a<0$ or $a>1$. Local stability requires $|f'(u_*)|<1$, and
$$
f'(0)=a,\qquad f'(u_\pm)=3-2a.
$$
Away from the boundary values:
* $a<-1$: all three fixed points are unstable.
* $-1<a<0$: $0$ is stable and $u_\pm$ are unstable.
* $0<a<1$: only $0$ exists, and it is stable.
* $1<a<2$: $0$ is unstable and $u_\pm$ are stable.
* $a>2$: all three are unstable.
At $a=-1,1,2$ a multiplier has modulus one and linear stability is inconclusive; at $a=0$, $0$ is strongly stable.
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