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Periodic orbits, localization in normal mode space, and the FermiPastaUlam problem
 

Summary: Periodic orbits, localization in normal mode space,
and the Fermi­Pasta­Ulam problem
S. Flacha
Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
M. V. Ivanchenko
Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom
O. I. Kanakov and K. G. Mishagin
Department of Radiophysics, Nizhny Novgorod University, Gagarin Avenue 23, 603950 Nizhny
Novgorod, Russia
Received 6 October 2007; accepted 10 November 2007
The Fermi­Pasta­Ulam problem was one of the first computational experiments. It has stirred the
physics community since, and resisted a simple solution for half a century. The combination of
straightforward simulations, efficient computational schemes for finding periodic orbits, and
analytical estimates allows us to achieve significant progress. Recent results on q-breathers, which
are time-periodic solutions that are localized in the space of normal modes of a lattice and maximize
the energy at a certain mode number, are discussed, together with their relation to the Fermi­Pasta­
Ulam problem. The localization properties of a q-breather are characterized by intensive parameters,
that is, energy densities and wave numbers. By using scaling arguments, q-breather solutions are
constructed in systems of arbitrarily large size. Frequency resonances in certain regions of wave
number space lead to the complete delocalization of q-breathers. The relation of these features to the

  

Source: Aickelin, Uwe - School of Computer Science, University of Nottingham
Flach, Sergej - Max-Planck-Institut für Physik komplexer Systeme

 

Collections: Computer Technologies and Information Sciences; Physics