# Computation and optimal perturbation of finite-time coherent sets for aperiodic flows without trajectory integration

## TL;DR

This paper introduces a novel method to compute and optimize finite-time coherent sets in aperiodic flows without trajectory integration, using transfer operators and spectral optimization to control mixing.

## Contribution

It develops a space-time generator approach for coherent set computation and provides explicit solutions for perturbation-based optimization of flow coherence.

## Key findings

- Avoids trajectory integration by using space-time generators
- Provides explicit solutions for flow perturbation optimization
- Demonstrates control of mixing through small velocity field changes

## Abstract

Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition of the state space into coherent sets is a popular way to reveal this essential macroscopic evolution. To compute coherent sets from an aperiodic time-dependent dynamical system we consider the relevant transfer operators and their infinitesimal generators on an augmented space-time manifold. This space-time generator approach avoids trajectory integration, and creates a convenient linearization of the aperiodic evolution. This linearization can be further exploited to create a simple and effective spectral optimization methodology for diminishing or enhancing coherence. We obtain explicit solutions for these optimization problems using Lagrange multipliers and illustrate this technique by increasing and decreasing mixing of spatial regions through small velocity field perturbations.