IFS Seminar

Abstract: Many important qualities of stellarators can be determined via the Poincar

A© plot of a symplectic return map. These qualities include the locations of the magnetic core (good for confinement),magnetic islands and chaotic regions (often bad for confinement, but

necessary for particle exhaust). In this talk, we present two methods for quickly and automatically analyzing the integrability of symplectic

maps.

First, we propose a kernel-based method for learning a single label-

ing function that is approximately invariant under the symplectic map.

From the labeling function, we can approximately recover the loca-

tions of invariant circles, islands, and chaos with few evaluations of the

underlying symplectic map. Additionally, the labeling function comes

with a residual, which serves as a measure of how invariant the label-

ing function is, and therefore as an indirect measure of chaos and map

complexity.

Second, we show how a modified version of the reduced rank extrapo-

lation method (named Birkhoff RRE) can be used to find an optimal

linear model of a trajectory using a single linear least-squares solve. Us-

ing the model, we classify trajectories as integrable or chaotic with few

iterations of the map. Furthermore, for the islands and invariant cir-

cles, a subsequent eigenvalue problem gives the number of islands and

the rotation number. Using these numbers, we directly find Fourier

parameterizations of invariant circles and islands. For both methods,

we show examples on the standard map and a stellarator configuration.