Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 31E / b

The line $x=0$ is invariant for every $a$; on it, $\dot{y}=1-y^2$, so the segment from $P_{-}$ to $P_{+}$ is a heteroclinic orbit. For $a=0$, the zero-energy set isThese separatrices divide the two families of closed orbits around the centers from the exterior trajectories. For $a=1/2$, both axial equilibria are saddles and $Q_{\pm }=(\pm \sqrt{3}/2,1/2)$ are stable foci. For $a=3/2$, the off-axis equilibria have disappeared, $P_{+}$ is a stable node, and $P_{-}$ remains a saddle.

The arrows, fixed-point types, invariant axis, and the exact $a=0$ separatrix are shown in the three phase portraits.