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 Hénon-Heiles system and the multidimensional $β$-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 $τ\lesssim 1$, 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.
Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method
On the behavior of the Generalized Alignment Index (GALI) method for regular motion in multidimensional Hamiltonian systems
研究了多维Hamilton系统正则轨道的$k$阶广义排列指数（GALI $_k$）的行为. GALI $_k$是一种有效的混沌指示器，当$2\leq k \leq N$，其中$N$是运动发生的环面的维数时，其对于规则运动渐近地达到正值。通过对Fermi-Pasta-Ulam-Tsingou（FPUT）$β$模型的两个典型的简单稳定周期轨道附近的几个正则轨道的研究，我们证明了当系统自由度为不同值时，当系统自由度的阶数$k$增加时，当系统的能量接近周期轨道的失稳能量时，系统的稳定岛消失，系统的渐近GALI $_k$值减小，而对于固定能量，当所考虑的规则轨道进一步远离周期轨道时，它们增加。此外，进行广泛的数值模拟，我们表明，该指数的行为并不依赖于其评估所需的初始偏差向量的选择。
Recurrence time entropy and weak chaos in Hamiltonian flows
São Paulo State University (UNESP)（圣保罗州立大学）；Institute of Geosciences and Exact Sciences（地球科学与精确科学研究所）；University of São Paulo（圣保罗大学）；Institute of Physics（物理研究所）；State University of Ponta Grossa（蓬塔格罗萨州立大学）
