Stokes sets are surfaces (codimension one) in space, across which or acquires an exponentially-small asymptotic contribution (in ), associated with a complex critical point of or . The Stokes sets are defined by the exponential dominance condition:
In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise.
The Stokes set consists of the rays in the complex -plane.
The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set:
The Stokes set takes different forms for , , and .
For , the set consists of the two curves
where are the two smallest positive roots of the equation
For , the Stokes set is expressed in terms of scaled coordinates
where satisfies the equation
For , there are two solutions , provided that . They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
For the Stokes set has two sheets. The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9). The second sheet corresponds to and it intersects the bifurcation set (§36.4) smoothly along the line generated by , . For the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation
This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the -axis by . One of the sheets is symmetrical under reflection in the plane , and is given by
Here is the root of the equation
and such that
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. With coordinates
the intersection lines with the bifurcation set are generated by , . Define
When the Stokes set is given by
where is the root of the equation
such that . This part of the Stokes set connects two complex saddles.
where is the positive root of the equation
In Figures 36.5.1–36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets). Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions.