A patch in time saves nine: Methods for the identification of localised dynamical behaviour and lifespans of coherent structures

TL;DR

This paper introduces a transfer operator-based method to identify and analyze the lifespan of localized coherent structures in dynamical systems, applicable to both simple and complex models without requiring full system knowledge.

Contribution

The paper presents a novel localised transfer operator approach for detecting and characterizing transient coherent structures and their lifespans in dynamical systems.

Findings

Effective in simple double well potential model

Applicable to complex rotating Boussinesq equations

Enables exploration of system properties without full dynamics knowledge

Abstract

We develop a transfer operator-based method for the detection of coherent structures and their associated lifespans. Characterising the lifespan of coherent structures allows us to identify dynamically meaningful time windows, which may be associated with transient coherent structures in the localised phase space, as well as with time intervals within which these structures experience fundamental changes, such as merging or separation events. The localised transfer operator approach we pursue allows one to explore the fundamental properties of a dynamical system without full knowledge of the dynamics. The algorithms we develop prove useful not only in the simple case of a periodically driven double well potential model, but also in more complex cases generated using the rotating Boussinesq equations.