Relatively Coherent Sets as a Hierarchical Partition Method

TL;DR

This paper introduces a hierarchical method for identifying relatively coherent sets in dynamical systems, extending existing finite-time coherence concepts to better analyze complex, nonautonomous flows with practical applications and computational considerations.

Contribution

It generalizes finite-time coherent sets to a hierarchical framework based on relative measures, enabling multi-scale analysis of complex dynamical systems.

Findings

Demonstrated the method on nonautonomous double gyre and standard map.

Applied the approach to empirical data from the 2010 Mexico Gulf oil spill.

Analyzed the computational complexity of the matrix estimation process.

Abstract

Finite time coherent sets [8] have recently been defined by a measure based objective function describing the degree that sets hold together, along with a Frobenius-Perron transfer operator method to produce optimally coherent sets. Here we present an extension to generalize the concept to hierarchially defined relatively coherent sets based on adjusting the finite time coherent sets to use relative mesure restricted to sets which are developed iteratively and hierarchically in a tree of partitions. Several examples help clarify the meaning and expectation of the techniques, as they are the nonautonomous double gyre, the standard map, an idealized stratospheric flow, and empirical data from the Mexico Gulf during the 2010 oil spill. Also for sake of analysis of computational complexity, we include an appendic concerning the computational complexity of developing the Ulam-Galerkin matrix extimates of the Frobenius-Perron operator centrally used here.