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

Article

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 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.

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References

  1. B.V. Chirikov, Phys. Rep. 180, 179 (1979)Google Scholar
  2. R.S. MacKay, J.D. Meiss, Hamiltonian Dynamical Systems (Adam Hilger, Bristol, 1987)Google Scholar
  3. M.A. Lieberman, A.J. Lichtenberg, Regular and Chaotic Dynamics (Springer Verlag, 1992)Google Scholar
  4. G. Contopoulos, Order and Chaos in Dynamical Astronomy (Springer, Berlin, 2002)Google Scholar
  5. Nonlinear Problems in Future Particle Accelerators, edited by W. Scandale, G. Turchetti (World Scientific, Singapore, 1991); T. Bountis, Ch. Skokos, Nucl. Instrum. Meth. Res. A 561, 173 (2006)Google Scholar
  6. S.C. Farantos, Z.-W. Qu, H. Zhu, R. Scinke, Int. J. Bifur. Chaos. 16, 1913 (2006)Google Scholar
  7. C. Jung, H.S. Taylor, E.L. Sibert, J. Phys. Chem. A 110, 5317 (2006)Google Scholar
  8. S. Flach, M.V. Ivanchenko, O.I. Kanakov, Phys. Rev. Lett. 95, 064102 (2005)Google Scholar
  9. J. Ford, Phys. Rep. 213, 271 (1992); G.P. Berman, F.M. Izrailev, Chaos 15, 015104 (2005)Google Scholar
  10. G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Meccanica 15, 9 (1980)Google Scholar
  11. G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Meccanica 15, 21 (1980)Google Scholar
  12. Ch. Skokos, T.C. Bountis, Ch. Antonopoulos, Physica D 231, 30 (2007)Google Scholar
  13. Ch. Skokos, J. Phys. A 34, 10029 (2001); Ch. Skokos, Ch. Antonopoulos, T.C. Bountis, M.N. Vrahatis, Prog. Theor. Phys. Supp. 150, 439 (2003); Ch. Skokos, Ch. Antonopoulos, T.C. Bountis, M.N. Vrahatis, J. Phys. A 37, 6269 (2004)Google Scholar
  14. F.F. Karney, Physica D 8, 360 (1983); J.D. Meiss, E. Ott, Phys. Rev. Lett. 55, 2741 (1985); V. Afraimovich, G.M. Zaslavsky, Lect. Notes Phys. 511, 59 (1998); Ch. Efthymiopoulos, G. Contopoulos, N. Voglis, R. Dvorak, J. Phys. A 30, 8167 (1997); R. Dvorak, G. Contopoulos, Ch. Efthymiopoulos, N. Voglis, Planet. Space Sci. 46, 1567 (1998)Google Scholar
  15. M. Spivak, Comprehensive Introduction to Differential Geometry, Vol. 1 (Publ. or Per. Inc., 1999)Google Scholar
  16. Ch. Antonopoulos, T. Bountis, Romai J. 2, 1 (2006); see also Ch. Antonopoulos, Ph.D. Thesis, Department of Mathematics, University of Patras, 2007Google Scholar
  17. J.H. Hubbard, B.B. Hubbard, Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach, Chapter 5 (Prentice Hall, 1999)Google Scholar
  18. N. Bourbaki, Éléments de mathématique, Livre II: Algèbre, Chapitre 3 (Hermann, 1958)Google Scholar
  19. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Chapter 2 (Cambridge University Press, 2003)Google Scholar
  20. R.I. McLachlan, G.R.W. Quispel, J. Phys. A 39, 5251 (2006)Google Scholar
  21. P.J. Prince, J.R. Dormand, J. Comp. Appl. Math. 7, 67 (1981)Google Scholar
  22. E. Christodoulidi, T. Bountis, Romai J. 2, 37 (2006)Google Scholar
  23. O. Merlo, L. Benet, Cel. Mech. Dyn. Astr. 97, 49 (2007)Google Scholar

Copyright information

© 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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