Periodic Orbits in Fermi-Pasta-Ulam-Tsingou Systems

Abstract

The FPUT paradox is the phenomenon whereby a one-dimensional chain of oscillators with nonlinear couplings shows non-ergodic behavior. The trajectory of the system in phase space, with a long wavelength initial condition, closely follows that of the Toda model over short times, as both systems seem to relax quickly to a non-thermal, metastable state. Over longer times, resonances in the FPUT spectrum drive the system towards equilibrium, away from the Toda trajectory. Similar resonances are observed in $q$-breather spectra, suggesting that $q$-breathers are involved in the route towards thermalization. In this article we investigate such resonances and show that they occur due to exact overlaps of $q$-breather frequencies of the type $m\Omega_1 = \Omega_k$. The resonances appear as peaks in the energy spectrum. Further, they give rise to new composite periodic orbits, which exist simultaneously with the original $q$-breathers. We find that such resonances are absent in integrable systems, as a consequence of the (infinite number of) conservation laws associated with integrability.