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

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.

References in corpus (8)

• Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks
• On the numerical approximation of the Perron-Frobenius and Koopman operator
• A Spectral Clustering Approach to Lagrangian Vortex Detection
• On the Lagrangian Dynamics of Atmospheric Zonal Jets and the Permeability of the Stratospheric Polar Vortex
• Flow networks: A characterization of geophysical fluid transport
• Geometry of the ergodic quotient reveals coherent structures in flows
• Estimating long-term behavior of periodically driven flows without trajectory integration
• Computing Coherent Sets using the Fokker-Planck Equation

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