# Trajectory-free approximation of phase space structures using the trajectory divergence rate

## TL;DR

This paper introduces the trajectory divergence rate, a new scalar field for analyzing local attraction or repulsion in phase space, aiding in the rapid approximation of invariant manifolds and coherent structures in dynamical systems.

## Contribution

It presents the derivation, properties, and applications of the trajectory divergence rate and ratio, extending their use to higher-dimensional systems.

## Key findings

- Effective identification of invariant manifolds
- Application to various example systems
- Extension to higher dimensions

## Abstract

This paper introduces the trajectory divergence rate, a scalar field which locally gives the instantaneous attraction or repulsion of adjacent trajectories. This scalar field may be used to find highly attracting or repelling invariant manifolds, such as slow manifolds, to rapidly approximating hyperbolic Lagrangian coherent structures, or to provide the local stability of invariant manifolds. This work presents the derivation of the trajectory divergence rate and the related trajectory divergence ratio for 2-dimensional systems, investigates their properties, shows their application to several example systems, and presents their extension to higher dimensions.