Why politics makes strange bedfellows: Dynamic model with DNS curves

We analyze a two-dimensional system of political behavior which has three equilibria in the uncontrolled version. After adding a control variable, two more equilibria occur and Skiba curves (also called DNS curves) can be analyzed. In this model, it is possible to derive under what conditions each of the different equilibria is a saddle point, a node, or a focus. In particular, for certain parameter ranges, all five equilibria have real eigenvalues. In this case, the Skiba curves can be computed in a more straightforward way than usual. The curves spiral outward, so any ray extending from the origin crosses these curves arbitrarily many times, as it alternately crosses regions for which it is optimal to approach each of the three equilibria.