Classification of symmetric attractors based on optimal transport theory

Symmetric dynamical systems can show complex phenomena that only reveal their symmetries (either the full symmetry group of the system, a smaller subgroup, or no symmetry) when considering long-time trajectories. Identifying the subgroup that characterizes the symmetry of the attractor is relevant for situations where the structure of the chaotic attractor suffers qualitative changes, as in crises and other bifurcations. Although there are well-established methods for the classification, most notably the "symmetry detectives" developed by Barany et al. [Phys. D: Nonlinear Phenom. 67(1), 66-87 (1993)], recent advances in machine learning and data-driven modeling offer promising new approaches to this problem. In this article, we will show how the theory and algorithms of optimal transport can efficiently quantify the distance between an invariant measure (either a long sequence of points or a multi-dimensional histogram) and its transformations under the actions of the group and assess the presence of a given subgroup. We show the potential of the new method using systems of coupled oscillators that are either fully symmetric or composed of symmetric subpopulations.