Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets

One aspect of the coexistence of regular structures and chaos in many dynamical systems is the emergence of coherent sets: If we place a large number of passive tracers in a coherent set at some initial time, then macroscopically they perform a collective motion and stay close together for a long period of time, while their surrounding can mix chaotically. Natural examples are moving vortices in atmospheric or oceanographic flows. In this article, we propose a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. This is done by constructing a random walk on the data points that captures both the inherent time-ordering of the data and the idea of closeness in space, which is at the heart of coherence. In the rich data limit, we can show equivalence to the well-established functional-analytic framework of coherent sets. One output of our method are "dynamical coordinates,"which reveal the intrinsic low-dimensional transport-based organization of the data.