Trajectory arclength reveals chaos

TL;DR

This paper introduces a new, simple chaos indicator based on phase space arclength and spectral analysis, validated on classical Hamiltonian systems, which is computationally efficient and suitable for high-dimensional systems.

Contribution

It presents a novel chaos detection method using phase space arclength and spectral density, avoiding complex computations and enabling large-scale analysis.

Findings

Distinguishes regular and chaotic motion via spectral density patterns.

Validated on Hénon-Heiles and Fermi-Pasta-Ulam systems.

Efficiently classifies initial conditions for chaos detection.

Abstract

In this paper we demonstrate that the phase space arclength of a trajectory, quantified by the time-averaged Lagrangian descriptor, is a robust and self-contained chaos indicator. By invoking Birkhoff's Ergodic Partition Theorem, we show that this scalar function distinguishes dynamical regimes via its power spectral density: for regular motion it converges to a delta function, whereas for chaotic trajectories the spectrum exhibits an inverse power-law $(1/\omega)$ driven by the phenomenon of dynamical stickiness. With this approach, we avoid the computation and simulation of the variational equations and the usage of neighboring orbits, making it the simplest geometrical chaos indicator derivable from Lagrangian descriptors. Its computational efficiency enables the study of high-dimensional systems and allows the generation of large datasets of classified initial conditions, ideal for training Machine Learning models. We validate these findings using the H\'enon-Heiles and the Fermi-Pasta-Ulam systems. By linking the geometrical properties of phase space to spectral analysis, this work provides the mathematical justification to establish Lagrangian descriptors as a rigorous, self-sufficient framework for the global analysis of chaos and regularity in Hamiltonian systems.