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

TL;DR

This paper introduces ΔL, a chaos indicator derived from the difference between Lagrangian descriptors of neighboring trajectories, calculable with a single additional integration and no variational equations. The authors derive its theoretical behavior, showing linear growth for regular orbits and exponential growth for chaotic ones, and define practical bounds. Through benchmarks on the Henon–Heiles system and the Chirikov Standard Map, ΔL achieves accuracy comparable to established LD-based indicators and SALI, while offering substantial computational efficiency. The work provides a scalable, accessible tool for chaos detection and phase-space analysis, with promising potential for higher-dimensional dynamical systems.

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 Hénon-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.

Figures (6)

• Figure 1: Histograms for the different chaos indicators considered in this analysis for the Hénon-Heiles hamiltonian calculated with $10^{5}$ randomly generated initial conditions and $\mathcal{H} = 1/8$ integrated for $\tau = 10^{4}$ units of time and $\| \boldsymbol{\beta} \| = \sigma = 10^{-8}$. A) distribution of the $\log_{10}(\Delta \mathcal{L})$ indicator; B) distribution of the $\log_{10}(\text{SALI})$ indicator; C) distribution of the $\log_{10}(\mathcal{D})$; D) distribution of the $\log_{10}(\mathcal{R})$; E) distribution of the $\log_{10}(\mathcal{C})$; F) distribution of the $\log_{10}(\mathcal{S})$. Note that the indicators given in Eq. \ref{['eq:chaos_inds']} were calculated with $n = 2$.
• Figure 2: Histograms for the different chaos indicators considered in this analysis for the Chirikov Standard Map calculated with a regular grid of size $320 \times 320$ initial conditions and $\mathrm{K} = 0.5 , 0.971635 \, \text{and} \, 1.5$ iterated for $N = 10^{5}$ iterations and $\| \boldsymbol{\beta} \| = \sigma = 10^{-8}$. A) distribution of the $\log_{10}(\Delta \mathcal{L})$ indicator; B) distribution of the $\log_{10}(\text{SALI})$ indicator; C) distribution of the $\log_{10}(\mathcal{D})$; D) distribution of the $\log_{10}(\mathcal{R})$; E) distribution of the $\log_{10}(\mathcal{C})$; F) distribution of the $\log_{10}(\mathcal{S})$. Note that the indicators given in Eq. \ref{['eq:chaos_inds']} were calculated with $n = 2$.
• Figure 3: A) Comparison of the time evolution of the $\Delta \mathcal{L}$ indicator as a function of the integration time $\tau$ for a regular (blue) and a chaotic (orange) initial condition in the Hénon--Heiles system with $\mathcal{H} = 1/8$. B) Comparison of the time evolution of the time-averaged $\Delta \mathcal{L}$ for the same initial conditions. As expected, for the regular initial condition the value is smaller than for the chaotic one and is approximately constant, while the evolution in the case of the chaotic trajectory is clearly different from a constant. The regular initial condition corresponds to $x = 0$, $y = 0.2$, and $p_y = 0$, while the chaotic one is given by $x = 0$, $y = -0.175$, and $p_y = 0$. To compute the neighboring trajectory, a separation of $\| \boldsymbol{\beta} \| = 10^{-8}$ was used for both cases.
• Figure 4: Comparison of orbit classification performance between $\Delta \mathcal{L}$ and SALI indicators for the Hénon-Heiles system for various energy values. A), C), E) and G) are the confusion matrices comparing the classifications obtained with $\Delta \mathcal{L}$ and SALI for the energies $\mathcal{H} = 1/8$, $1/10$, $1/12$ and $1/15$. Here, True label stands for the classification provided by SALI while Predicted label refers to the classification obtained with $\Delta \mathcal{L}$. B), D), F) and H) are the corresponding Poincaré sections. For each section, $10^{5}$ randomly generated initial conditions classified with the $\Delta \mathcal{L}$ indicator, integrated for $\tau = 10^{4}$ units of time and $\|\boldsymbol{\beta}\| = 10^{-8}$, are overlaid. Regular orbits are depicted in blue and chaotic orbits in red.
• Figure 5: Comparison of orbit classification performance between $\Delta \mathcal{L}$ and SALI indicators for the Chirikov Standard Map at different $\mathrm{K}$ values. Figures A), C) and E) are the confusion matrices comparing the classification obtained with both indicators for $\mathrm{K} = 0.5$, $0.971635$ and $1.5$. Here, True label stands for the classification provided by SALI while Predicted label refers to the classification obtained with $\Delta \mathcal{L}$. B), D) and F) are the corresponding Poincaré sections for those $\mathrm{K}$ values. For each section, we overlay a uniform $320\times 320$ grid of initial conditions, iterated for $10^{5}$ steps and classified via the $\Delta\mathcal{L}$ indicator with $\lVert\boldsymbol{\beta}\rVert = 10^{-8}$. The ones shown in blue are classified as regular and the ones in red as chaotic.