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

We present a thorough analysis of computing the Generalized Alignment Index (GALI), a rapid and effective chaos indicator, through a simple multi-particle approach, which avoids the use of variational equations. We develop a theoretical leading-order estimation of the error in the computed GALI for both the variational method (VM) and the multi-particle method (MPM), and confirm its predictions through extensive numerical simulations of two well-known Hamiltonian models: the H\'enon-Heiles and the $\beta$-Fermi-Pasta-Ulam-Tsingou systems. For these models the GALIs of several orders are computed and the MPM results are compared to the VM outcomes. The dependence of the accuracy of the MPM on the renormalization time, integration time step, as well as the deviation vector size, is studied in detail. We find that the implementation if the MPM in double machine precision ($\varepsilon \approx 10^{-16}$) is reliable for deviation vector magnitudes centred around $d_0\approx \varepsilon^{1/2}$, renormalization times $\tau \lesssim 1$, and relative energy errors $E_r \lesssim \varepsilon^{1/2}$. These results are valid for systems with many degrees of freedom and for several orders of the GALIs, with the MPM particularly capturing very accurately the $\textrm{GALI}_2$ behavior. Our results show that the computation of the GALIs by the MPM is a robust and efficient method for investigating the global chaotic dynamics of autonomous Hamiltonian systems, something which is of distinct importance in cases where it is difficult to explicitly write the system's variational equation or when these equations are too cumbersome.