Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 31A / d / Solution

For $\nu =\mu +1>0$ small, $(1,0)$ has two negative eigenvalues and is a stable node. The two nearby equilibria satisfy $u=\pm \sqrt{\nu }+O(\nu )$ and are saddles. Their stable and unstable separatrices form the local phase portrait; trajectories in the wedge between the stable manifolds approach $(1,0)$, while trajectories crossing an unstable direction leave the neighbourhood.