I Butterfly Points

There are two other simple cases where analytic roots of eq. (39) can be obtained. If we set w = 0, the resulting quadratic in v2 can be solved to give 

It is also possible to find solutions with e = 0. These are not solutions of eqs (28a)-(28e) in the strict sense nor are they physically meaningful, but they will be useful in several connections later. One such solution is w = y and v given by eq. (43) the other two sets are given by 

The solution (44a), (44b) turns out to be a limit point of eq. (39). While the solution given by eqs (45a) and (45b) is a regular point for eq. (39), it is worth noting that it only makes sense if y s 12 or y 4. A final qualitative result concerns no a solution, but the lack of one. A root of eq. (39) will have a vertical asymptote if the coefficient of w4 vanishes, which happens (for y = 3) if 

FIGURE 2 Butterfly candidates w and e vs. v under URP from Jorgensen et al. (1984). 

We turn now to the solutions of eq. (39). Using the methods of Golubitsky and Schaeffer (1979), one can show (Farr, 1986) that the solution of this equation (regarding v as our distinguished parameter) results in the four qualitatively different diagrams shown in Fig. 4. Actually the methods of the 

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