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swallowtail bifurcation set

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1: 36.4 Bifurcation Sets
K = 3 , swallowtail bifurcation set: …
§36.4(ii) Visualizations
See accompanying text
Figure 36.4.2: Bifurcation set of swallowtail catastrophe. Magnify
2: 36.5 Stokes Sets
They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set36.4). …
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
3: 36.11 Leading-Order Asymptotics
§36.11 Leading-Order Asymptotics
and far from the bifurcation set, the cuspoid canonical integrals are approximated by …
36.11.4 Ψ 3 ( x , 0 , 0 ) = 2 π ( 5 | x | 3 ) 1 / 8 { exp ( - 2 2 ( x / 5 ) 5 / 4 ) ( cos ( 2 2 ( x / 5 ) 5 / 4 - 1 8 π ) + o ( 1 ) ) , x + , cos ( 4 ( | x | / 5 ) 5 / 4 - 1 4 π ) + o ( 1 ) , x - .
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 + .
36.11.6 Ψ 3 ( 0 , 0 , z ) = Γ ( 1 3 ) | z | 1 / 3 3 + { o ( 1 ) , z + , 2 π 5 1 / 4 ( 3 | z | ) 3 / 4 ( cos ( 2 3 ( 3 | z | 5 ) 5 / 2 - 1 4 π ) + o ( 1 ) ) , z - .
4: 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 …The rings are almost circular (radii close to ( Δ x ) / 9 and varying by less than 1%), and almost flat (deviating from the planes z n by at most ( Δ z ) / 36 ). …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. …
§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals
The zeros of these functions are curves in x = ( x , y , z ) space; see Nye (2007) for Φ 3 and Nye (2006) for Φ ( H ) .