Trajectory arclength reveals chaos

TL;DR

This work tackles efficient chaos detection in Hamiltonian dynamics by introducing a time-averaged Lagrangian descriptor, $\Omega = \mathcal{L}_f(\mathbf{x}_0,t_0,\tau)/\tau$, whose spectral content distinguishes regular from chaotic motion through $\hat{\Omega}_{reg} = \xi\,\delta(\omega)$ and $\hat{\Omega}_{chaos} \sim \omega^{-1}$, grounded in Birkhoff's ergodic partition. The method avoids variational equations and neighboring trajectories, offering $\mathcal{O}(N)$ scalability and enabling large datasets for machine learning; it is validated on the Hénon–Heiles and Fermi–Pasta–Ulam models, with Welch-based PSD providing robust ensemble separation. The results establish a simple, self-contained geometric framework for global chaos analysis in Hamiltonian systems and highlight the practical benefits of LD-based indicators for high-dimensional dynamics. This approach opens pathways to studying complex phenomena like Arnold diffusion and to integrating chaos classification into scalable computational pipelines for physics and related fields.

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 $(1/ω)$ 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 Hénon-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.

Figures (13)

• Figure 1: Illustration of Birkhoff's Ergodic Partition Theorem using $\Omega$ in the Hénon-Heiles system. (A) Time evolution of $\Omega$ for an ensemble of $5000$ trajectories in the Hénon-Heiles system whose total energy is $\mathcal{H} = 1/8$. (B) Value of $\Omega$ for the same ensemble of initial conditions depicted in the Poincaré section given by $x = 0$ and $p_x > 0$ (shown in Fig. \ref{['fig:Poincare_sec_HH']}). Panel (A) shows that the chaotic initial conditions (according to the criteria given for $\Omega$) are confined between two regimes of regular trajectories. This is consistent with Birkhoff's Ergodic Partition Theorem for systems with $2$ degrees of freedom. Furthermore, the value of $\Omega$ effectively identifies the different structures present in the system's phase space, as it can be seen in panel (B).
• Figure 2: Dynamical stickiness in the Hénon-Heiles system. (A) Intersections of a chaotic trajectory with the Poincaré section defined by $x = 0$ and $p_x > 0$. The gray points belong to the chaotic sea, while the red points correspond to the sticky regime, where the trajectory lingers near the boundaries of a regular island. (B) and (C) Time series of $y(t)$ and $p_y(t)$ during the sticky interval. Note the quasi-regular oscillations observed within this interval before the trajectory diffuses back into the chaotic sea.
• Figure 3: Spectral signatures of dynamical stickiness. Fourier transform computed for (A) the coordinate $y(t)$ and (B) the momentum $p_y(t)$. The blue curves correspond to the spectral analysis of the trajectory during the sticky interval, as shown in Fig. \ref{['fig:sticky_1']}, while the orange curves represent the behavior in the chaotic sea. The comparison reveals that the motion during the sticky interval exhibits sharp spectral lines, in contrast with the broad spectrum characteristic of chaotic motion.
• Figure 4: Poincaré Section of the Hénon-Heiles System. Poincaré section defined by $x = 0$ and $p_x > 0$ for the Hénon-Heiles, given in Eq. \ref{['eq:Ham_HH']}, at a total energy $\mathcal{H} = 1/8$. A regular (blue) and a chaotic (orange) initial condition are overlaid, illustrating the mixed phase space structure. These two representative initial conditions were selected to validate the performance of the chaos indicator $\Omega$.
• Figure 5: Evolution of $\Delta \mathcal{L}$ and its time average in the Hénon-Heiles system. (A) Time evolution of the $\Delta \mathcal{L}$ indicator, defined in Eq. \ref{['eq:DL_def']}, for a regular (blue) and a chaotic (orange) initial condition of the Hénon-Heiles system computed with $\|\delta \mathbf{x}\| = 10^{-8}$. (B) Corresponding time evolution of the time-averaged $\Delta \mathcal{L}$ chaos indicator, computed under the same conditions, for the same two trajectories. (C) and (D) Analogous results computed with and initial separation $\|\delta \mathbf{x}\| = 10^{-12}$. Note that the regular trajectory exhibits linear growth, resulting in a constant average.