1 shows the stability diagram for the representative case where $K=5$. Here the various stability regions labeled A-E correspond to the phase portraits shown in Fig. 2.

We realize that these figures appear complicated at first glance, so let us begin by offering a few general remarks about them. Figure 1 is divided into five regions, A-E, by the bifurcation curves labeled saddle-node, Hopf, homoclinic, and SNIPER. In the places where two or more of these curves nearly coincide, Fig. 1(a) becomes especially confusing. To clarify what is going on in such regions, Figs. 1(b) and 1(c) zoom in near two codimension-2 points of interest (to be discussed in detail later). Since even these figures can be hard to interpret, we have tried to make everything as clear as possible by presenting a schematic Fig. 1(d). Unlike Figs. 1(a)-(c), which are numerically accurate, Fig. 1(d) is only topologically correct. We have distorted some of stability regions and pulled certain curves apart to make the layout of the diagram transparent, and to highlight the three different codimension-2 points that will later be seen to organize the entire diagram.

A similar but incomplete version of Fig. 1 was obtained previously by Antonsen 