Abstract
Dynamical systems often exhibit the emergence of long-lived coherent sets,
which are regions in state space that keep their geometric integrity to a high
extent and thus play an important role in transport. In this article, we
provide a method for extracting coherent sets from possibly sparse Lagrangian
trajectory data. Our method can be seen as an extension of diffusion maps to
trajectory space, and it allows us to construct "dynamical coordinates" which
reveal the intrinsic low-dimensional organization of the data. The only a
priori knowledge about the dynamics that we require is a locally valid notion
of distance, which renders our method highly suitable for automated data
analysis. We show convergence of our method to the analytic transfer operator
framework of coherence in the infinite data limit, and illustrate its potential
on several two- and three-dimensional examples as well as real world data.
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