Three applications of Topological Data Analysis to chaotic magnetic fields

Three applications of Topological Data Analysis to the study of chaotic fields are developed. A procedure for the automatic classification of the orbits of magnetic field lines into topologically distinct classes using Vietoris-Rips persistent homology is presented and tested for a toy model of a perturbed tokamak. A method for estimating the distribution of the size of islands in the phase space of a Hamiltonian system or area-preserving map by sub-level set persistent homology is explored. This method is used to analyse the case of an accelerator mode island in the phase space of Chirikov's standard map and the possibility of detecting the self-similar island hierarchy responsible for anomalous transport in this model is investigated. Finally, it is suggested that TDA provides a toolset for the detection and characterisation of renormalisation group transformations which leave structures in Hamiltonian phase spaces invariant. The specific example of detecting the transform which leaves the neighborhood of a hyperbolic fixed point of the perturbed pendulum invariant is investigated using two different TDA approaches. Both of which are found to be partially sensitive to the symmetry in question but only weakly.