Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 31E / iii / Solution

Away from $x=\pm 1$, the sign of $\dot{x}=x^2-1$ is fixed within each of the three invariant strips, so $x(t)$ is strictly monotone. On $x=\pm 1$, the equation is $\dot{y}=\pm ky$, whose nonconstant solutions are exponential and hence not periodic. Therefore the monotone coordinate excludes periodic orbits, and there are