# Lagrangian Descriptors for Two Dimensional, Area Preserving, Autonomous and Nonautonomous Maps

## TL;DR

This paper extends Lagrangian descriptors to 2D area-preserving maps, both autonomous and nonautonomous, providing explicit formulas, rigorous connections to invariant manifolds, and computational analysis on chaotic saddles.

## Contribution

It generalizes Lagrangian descriptors to discrete 2D maps, offers explicit evaluations, and clarifies their relation to stable and unstable manifolds in autonomous and nonautonomous settings.

## Key findings

- Explicit formulas for Lagrangian descriptors in certain norms
- Rigorous link between singular sets and invariant manifolds
- Effective computation of chaotic saddles in Hénon maps

## Abstract

In this paper we generalize the method of Lagrangian descriptors to two dimensional, area preserving, autonomous and nonautonomous discrete time dynamical systems. We consider four generic model problems--a hyperbolic saddle point for a linear, area-preserving autonomous map, a hyperbolic saddle point for a nonlinear, area-preserving autonomous map, a hyperbolic saddle point for linear, area-preserving nonautonomous map, and a hyperbolic saddle point for nonlinear, area-preserving nonautonomous map. The discrete time setting allows us to evaluate the expression for the Lagrangian descriptors explicitly for a certain class of norms. This enables us to provide a rigorous setting for the notion that the "singular sets'' of the Lagrangian descriptors correspond to the stable and unstable manifolds of hyperbolic invariant sets, as well as to understand how this depends upon the particular norms that are used. Finally we analyze, from the computational point of view, the performance of this tool for general nonlinear maps, by computing the "chaotic saddle'' for autonomous and nonautonomous versions of the H\'enon map.