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elliptic umbilic bifurcation set

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1: 36.4 Bifurcation Sets
Elliptic umbilic bifurcation set (codimension three): for fixed z , the section of the bifurcation set is a three-cusped astroid …
§36.4(ii) Visualizations
See accompanying text
Figure 36.4.3: Bifurcation set of elliptic umbilic catastrophe. Magnify
2: 36.5 Stokes Sets
See accompanying text
Figure 36.5.8: Sheets of the Stokes surface for the elliptic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify
3: 36.7 Zeros
§36.7(iii) Elliptic Umbilic Canonical Integral
The zeros are lines in x = ( x , y , z ) space where ph Ψ ( E ) ( x ) is undetermined. 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 …Near z = z n , and for small x and y , the modulus | Ψ ( E ) ( x ) | has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose z and x repeat distances are given by …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. …
4: 36.11 Leading-Order Asymptotics
§36.11 Leading-Order Asymptotics
and far from the bifurcation set, the cuspoid canonical integrals are approximated by …
36.11.5 Ψ 3 ( 0 , y , 0 ) = Ψ 3 ( 0 , - y , 0 ) ¯ = exp ( 1 4 i π ) π / y ( 1 - ( i / 3 ) exp ( 3 2 i ( 2 y / 5 ) 5 / 3 ) + o ( 1 ) ) , y + .
36.11.7 Ψ ( E ) ( 0 , 0 , z ) = π z ( i + 3 exp ( 4 27 i z 3 ) + o ( 1 ) ) , z ± ,
36.11.8 Ψ ( H ) ( 0 , 0 , z ) = 2 π z ( 1 - i 3 exp ( 1 27 i z 3 ) + o ( 1 ) ) , z ± .