Periodic orbits, localization in normal mode space, and the Fermi–Pasta–Ulam problem

Abstract

The Fermi–Pasta–Ulam 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 Fermi–Pasta–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 Fermi–Pasta–Ulam problem are discussed.