# bifurcation sets

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## 6 matching pages

##### 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

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►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 $\mathbf{x}=\mathbf{0}$, the uniform asymptotic approximations of §36.12 can be used.
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##### 3: 36.5 Stokes Sets

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►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set:
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►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
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►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.
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►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles.
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##### 4: 36.14 Other Physical Applications

##### 5: 36.7 Zeros

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►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.
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