A simple chaos indicator based on the Lagrangian descriptor difference of neighboring orbits

TL;DR

This paper introduces a new, simple chaos indicator based on the difference in Lagrangian descriptors between neighboring orbits, which is easy to implement and effective in detecting chaos.

Contribution

The paper proposes a novel chaos indicator derived from Lagrangian descriptors that is simpler yet as accurate as existing complex methods.

Findings

The difference LD effectively detects chaos in benchmark systems.

It matches the accuracy of more elaborate chaos indicators.

The method is easy to implement and interpret.

Abstract

In this paper we introduce a chaos indicator derivable from Lagrangian descriptors (LDs), defined as the difference in LD values between two neighboring trajectories. This difference LD is remarkably easy to implement and interpret, offering a direct and intuitive measure of dynamical behavior. We provide a heuristic argument linking its growth to the regularity or chaoticity due to the underlying initial condition, considering the arclength-based formulation of LDs. To evaluate its effectiveness, we benchmark it against more elaborate LD-based chaos indicators and the Smaller Aligment Index (SALI) using two prototypical systems: the H\'enon-Heiles system and the Chirikov Standard Map. Our results show that, despite its simplicity, the difference LD matches the accuracy of more sophisticated methods, making it a robust and accessible tool for chaos detection in dynamical systems.