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

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.