Max Ruth (Affiliation TBA)

"Numerical Methods for Quantifying Integrability in Stellarators"

Abstract: Many important qualities of stellarators can be determined via the Poincaré 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 labeling function that is approximately invariant under the symplectic map. From the labeling function, we can approximately recover the locations 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 labeling function is, and therefore as an indirect measure of chaos and map complexity.

Second, we show how a modified version of the reduced rank extrapolation method (named Birkhoff RRE) can be used to find an optimal linear model of a trajectory using a single linear least-squares solve. Using the model, we classify trajectories as integrable or chaotic with few iterations of the map. Furthermore, for the islands and invariant circles, 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.