Lagrangian descriptors for open maps

TL;DR

This paper introduces a method using Lagrangian descriptors to analyze open maps, specifically revealing the structure of the chaotic repeller in the open tribaker map, with potential implications for chaotic scattering and semiclassical theories.

Contribution

It adapts Lagrangian descriptors to open maps, providing a simple way to identify the chaotic repeller and its homoclinic tangles in classical and quantum chaos systems.

Findings

Successfully identified the inner structure of the chaotic repeller.

Detected homoclinic tangles of periodic orbits.

Potential applications in chaotic scattering and semiclassical analysis.

Abstract

We adapt the concept of Lagrangian descriptors, which have been recently introduced as efficient indicators of phase space structures in chaotic systems, to unveil the key features of open maps. We apply them to the open tribaker map, a paradigmatic example not only in classical but also in quantum chaos. Our definition allows to identify in a very simple way the inner structure of the chaotic repeller, which is the fundamental invariant set that governs the dynamics of this system. The homoclinic tangles of periodic orbits (POs) that belong to this set are clearly found. This could also have important consequences for chaotic scattering and in the development of the semiclassical theory of short POs for open systems.