The European Physical Journal Special Topics

, Volume 165, Issue 1, pp 5–14 | Cite as

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



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 GALIk 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 GALIk 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.


Hamiltonian System Lyapunov Exponent Singular Value Decomposition European Physical Journal Special Topic Deviation Vector 
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© EDP Sciences and Springer 2008

Authors and Affiliations

  1. 1.Astronomie et Systèmes Dynamiques, IMCCE, Observatoire de Paris, 77 avenue Denfert-RochereauParisFrance
  2. 2.Department of Mathematics, Division of Applied Analysis and Center for Research and Applications of Nonlinear Systems (CRANS)University of PatrasPatrasGreece

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