On the Behavior of the Generalized Alignment Index (GALI) Method for Regular Motion in Multidimensional Hamiltonian Systems

Abstract

We investigate the behavior of the Generalized Alignment Index of order k (GALIk ) for regular orbits of multidimensional Hamiltonian systems. The GALIk is an efficient chaos indicator, which asymptotically attains positive values for regular motion when 2≤k ≤N, with N being the dimension (D) of the torus on which the motion occurs. By considering several regular orbits in the neighborhood of two typical simple, stable periodic orbits of the Fermi-Pasta-Ulam-Tsingou (FPUT) β model for various values of the system's degrees of freedom, we show that the asymptotic GALIk values decrease when the order k of the index increases and when the orbit's energy approaches the periodic orbit's destabilization energy where the stability island vanishes, while they increase when the considered regular orbit moves further away from the periodic one for a fixed energy. In addition, by performing extensive numerical simulations we show that the behavior of the index does not depend on the choice of the initial deviation vectors needed for its evaluation.

Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method

“…In our study we will use the GALI method, which was introduced in [74], and proved to be a very efficient chaos detection technique as it has been successfully used for studying the chaoticity of several dynamical systems, see e.g. [95,96,97,98,99,100]. According to [74] the GALI of order k (GALI k ), 2 ≤ k ≤ 2N, is defined to be the volume of the generalized parallelogram having as edges k normalized deviation vectors…”

4 Boyutlu Fermiyonik Modelde Kaosun Genelleştirilmiş Hizalama İndeksi Yöntemiyle İncelenmesi

“…Oysa N < k ≤ 2N lineer bağımsız başlangıç sapma vektörleri ile başlarsak, o zaman asimptotik GALI değeri sıfır olacaktır çünkü bazı sapma vektörleri sonunda lineer bağımlı hale gelecek ve yine hepsi torusun N boyutlu teğet vektör uzayına düşecektir. Bir N boyutlu torus üzerinde bulunan düzenli yörüngeler için 𝐺𝐴𝐿𝐼 𝑘 'ın genel davranışı (Moges, 2020) 𝐺𝐴𝐿𝐼 𝑘 (𝑡) ∝ 𝑒 −[(𝜎 1 −𝜎 2 )+(𝜎 1 −𝜎 3 )+⋯+(𝜎 1 −𝜎 𝑘 )]𝑡 (8) 2 ≤ k ≤ 2N ile birlikte k sapma vektörü 𝜔 ̂1, 𝜔 ̂2, . .…”