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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ia/paper-2/1b/solution