Butterfly singularity

The roots of eq. (39) with v and y considered as parameters determine branches of candidates for butterfly singularity points, which advance to that status if certain conditions are not violated. We have already seen how the conditions A 0, e 0, w —y arise naturally in the derivation of eq. (39) as guarantees that solutions of the latter equation are valid solutions of (28a)-(28e). A separate class of conditions arise from the theory of Golubitsky and Schaeffer, allowing one to relate the quasi-global behaviour of the system of interest to that of simple polynomial functions. For example, the minimum conditions from the theory for a butterfly point are that eqs (28a)-(28e) be satisfied and also... 

An example will help to clear this point up. Fixing the value of y at 15, we can compute the cusp of fifth-order (or butterfly) singular points originating from the sixth-order singularity in the feasible region. When these points are plotted in the v-B plane, the result is the upper graph in Fig. 13. Choosing v and B to have the values 0.7520 and 930,000, the hysteresis variety projected... 

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