I've recently started to study the root locus concept and I have got this question about the branches. Please refer to this problem here in the link:
In that example, it can be seen that the branches emerging from the open loop poles s=0 and s=-2 when k=0 move along the real axis towards each other as k increases up to the breakaway point. After this, when k is further increased, one of the branches moves downwards and the other moves upwards. It is seen in this case that the branch from the poles s=-2 moves downward and the branch from s=0 moves upward.
Now my question is what is the factor that decides the upward/downward of the branches. Please provide a logical proof.