Dresden 2009 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 27: Poster II

DY 27.21: Poster

Thursday, March 26, 2009, 16:00–18:00, P1B

Detecting chaos and determining the dimensions of tori in Fermi-Pasta-Ulam lattices by the Generalized Alignment Index method — •Charalampos Skokos¹, Tassos Bountis², and Chris Antonopoulos³ — ¹Max Planck Institute for the Physics of Complex Systems, Nothnitzer Str. 38, D-01187 Dresden, Germany — ²Department of Mathematics, University of Patras, GR-26500, Patras, Greece — ³Universite Libre de Bruxelles, Campus de la Plaine, CP 231, Boulevard du Triomphe, B-1050, Brussels, Belgium

The recently introduced Generalized Alignment Index (GALI) method of chaos detection (Skokos et al. Physica D, 2007, 231, 30) is applied to distinguish efficiently between regular and chaotic orbits of multi–dimensional Hamiltonian systems. The GALI of order k (GALI_k) is proportional to volume elements formed by k initially linearly independent unit deviation vectors whose magnitude is normalized to unity from time to time. For chaotic orbits, GALI_k tends exponentially to zero with exponents that involve the values of several Lyapunov exponents, while in the case of regular orbits, GALI_k fluctuates around nonzero values or goes to zero following particular power laws that depend on the dimension of the torus and on the order k. We apply these indices to rapidly detect chaotic oscillations, identify low–dimensional tori of Fermi-Pasta-Ulam (FPU) lattices and predict weak diffusion away from quasiperiodic motion, long before it is actually observed in the oscillations (Skokos et al. Eur. Phys. J. Special Topics, 2008, 165, 5). We also present an efficient computation scheme of the GALI’s, based on the Singular Value Decomposition (SVD) algorithm.