# Linear response for the dynamic Laplacian and finite-time coherent sets

## TL;DR

This paper develops a linear response theory for the eigenfunctions of the dynamic Laplace operator to understand how finite-time coherent sets in nonlinear dynamical systems change under small parameter variations, with efficient numerical methods demonstrated.

## Contribution

It introduces a novel linear response framework for the dynamic Laplace operator's eigenfunctions, enabling analysis of coherent set sensitivity to parameter changes.

## Key findings

- Linear response theory for dynamic Laplace eigenfunctions established.
- Numerical methods based on finite-element approach developed.
- Numerical examples demonstrate practical applicability.

## Abstract

Finite-time coherent sets represent minimally mixing objects in general nonlinear dynamics, and are spatially mobile features that are the most predictable in the medium term. When the dynamical system is subjected to small parameter change, one can ask about the rate of change of (i) the location and shape of the coherent sets, and (ii) the mixing properties (how much more or less mixing), with respect to the parameter. We answer these questions by developing linear response theory for the eigenfunctions of the dynamic Laplace operator, from which one readily obtains the linear response of the corresponding coherent sets. We construct efficient numerical methods based on a recent finite-element approach and provide numerical examples.