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bifurcation sets


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1: 36.4 Bifurcation Sets
§36.4 Bifurcation Sets
K = 1 , fold bifurcation set: … K = 2 , cusp bifurcation set: … K = 3 , swallowtail bifurcation set: …
§36.4(ii) Visualizations
2: 36.15 Methods of Computation
Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from x = 0 , the uniform asymptotic approximations of §36.12 can be used. …
3: 36.5 Stokes Sets
The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set36.4). … This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. … In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. …
See accompanying text
Figure 36.5.7: Sheets of the Stokes surface for the swallowtail catastrophe (colored and with mesh) and the bifurcation set (gray). … Magnify
4: 36.14 Other Physical Applications
The physical manifestations of bifurcation sets are caustics. …
5: 36.7 Zeros
Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the z -axis that is far from the origin, the zero contours form an array of rings close to the planes …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. …
6: 36.11 Leading-Order Asymptotics
and far from the bifurcation set, the cuspoid canonical integrals are approximated by …