# An Extension of Discrete Lagrangian Descriptors for Unbounded Maps

## TL;DR

This paper extends Discrete Lagrangian Descriptors to unbounded maps by incorporating an escape time criterion, enabling detailed phase space analysis of complex dynamical systems like the Hénon map.

## Contribution

It introduces a novel extension of Discrete Lagrangian Descriptors that handles unbounded maps using escape times, enhancing phase space exploration capabilities.

## Key findings

- Successfully visualized invariant manifolds and chaotic structures in the Hénon map.
- Demonstrated the method's ability to reveal intricate phase space features.
- Extended the applicability of Lagrangian Descriptors to unbounded dynamical systems.

## Abstract

In this paper we provide an extension for the method of Discrete Lagrangian Descriptors with the purpose of exploring the phase space of unbounded maps. The key idea is to construct a working definition, that builds on the original approach introduced in Lopesino et al. (2015), and which relies on stopping the iteration of initial conditions when their orbits leave a certain region in the plane. This criterion is partly inspired by the classical analysis used in Dynamical Systems Theory to study the dynamics of maps by means of escape time plots. We illustrate the capability of this technique to reveal the geometrical template of stable and unstable invariant manifolds in phase space, and also the intricate structure of chaotic sets and strange attractors, by applying it to unveil the phase space of a well-known discrete time system, the H\'enon map.