Eur. Phys. J. Special Topics 165, 5-14 (2008)
https://doi.org/10.1140/epjst/e2008-00844-2

Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi–Pasta–Ulam lattices by the Generalized Alignment Index method

¹ Astronomie et Systèmes Dynamiques, IMCCE, Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris, France
² Department of Mathematics, Division of Applied Analysis and Center for Research and Applications of Nonlinear Systems (CRANS), University of Patras, 26500 Patras, Greece

Corresponding authors: hskokos@imcce.fr bountis@math.upatras.gr antonop@math.upatras.gr

Abstract

The recently introduced GALI method is used for rapidly detecting chaos, determining the dimensionality of regular motion and predicting slow diffusion in multi-dimensional Hamiltonian systems. We propose an efficient computation of the GALI_k indices, which represent volume elements of k randomly chosen deviation vectors from a given orbit, based on the Singular Value Decomposition (SVD) algorithm. We obtain theoretically and verify numerically asymptotic estimates of GALIs long-time behavior in the case of regular orbits lying on low-dimensional tori. The GALI_k indices are applied to rapidly detect chaotic oscillations, identify low-dimensional tori of Fermi–Pasta–Ulam (FPU) lattices at low energies and predict weak diffusion away from quasiperiodic motion, long before it is actually observed in the oscillations.