Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 2 / 1B / Solution

Let $f(u)=au(1-u^2)$. A fixed point satisfies $u=f(u)$, soThe nonzero fixed points exist for $a<0$ or $a>1$. Local stability requires $|f^{\prime }(u_{\ast })|<1$, andAway from the boundary values:

At $a=-1,1,2$ a multiplier has modulus one and linear stability is inconclusive; at $a=0$, $0$ is strongly stable.