Efficient detection of chaos through the computation of the Generalized Alignment Index (GALI) by the multi-particle method

TL;DR

This paper introduces a multi-particle method for efficiently computing the GALI chaos indicator, eliminating the need for variational equations, and validates its robustness and accuracy through extensive numerical simulations on prototypical Hamiltonian systems.

Contribution

The paper presents a novel multi-particle approach for GALI computation that is robust, accurate, and applicable to high-dimensional Hamiltonian systems, avoiding variational equations.

Findings

The multi-particle method performs reliably with deviation vector sizes around the square root of machine precision.

It remains accurate for renormalization times up to approximately one.

The method is effective for systems with many degrees of freedom, demonstrating robustness and efficiency.

Abstract

We present a method for the computation of the Generalized Alignment Index (GALI), a fast and effective chaos indicator, using a multi-particle approach that avoids variational equations. We show that this approach is robust and accurate by deriving a leading-order error estimation for both the variational (VM) and the multi-particle (MPM) methods, which we validate by performing extensive numerical simulations on two prototypical models: the two degrees of freedom H\'enon-Heiles system and the multidimensional $\beta$-Fermi-Pasta-Ulam-Tsingou chain of oscillators. The dependence of the accuracy of the GALI on control parameters such as the renormalization time, the integration time step and the deviation vector size is studied in detail. We test the MPM implemented with double precision accuracy ($\varepsilon \approx 10^{-16}$) and find that it performs reliably for deviation vector sizes $d_0\approx \varepsilon^{1/2}$, renormalization times $\tau \lesssim 1$, and relative energy errors $E_r \lesssim \varepsilon^{1/2}$. These results hold for systems with many degrees of freedom and demonstrate that the MPM is a robust and efficient method for studying the chaotic dynamics of Hamiltonian systems. Our work makes it possible to explore chaotic dynamics with the GALI in a vast number of systems by eliminating the need for variational equations.