Periodic orbits in Fermi–Pasta–Ulam–Tsingou systems

The Fermi–Pasta–Ulam–Tsingou (FPUT) paradox is the phenomenon whereby a one-dimensional chain of oscillators with nonlinear couplings shows long-lived nonergodic behavior prior to thermalization. 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 toward equilibrium, away from the Toda trajectory. Similar resonances are observed in $q$-breather spectra, suggesting that $q$-breathers are involved in the route toward thermalization. In this article, we first review previous important results related to the metastable state, solitons, and $q$-breathers. We then investigate orbit bifurcations of $q$-breathers and show that they occur due to resonances, where the $q$-breather frequencies become commensurate as $m Ω 1= Ω k$⁠. The resonances appear as peaks in the breather energy spectrum. Furthermore, they give rise to new “composite periodic orbits,” which are nonlinear combinations of multiple $q$-breathers that exist following orbit bifurcations. We find that such resonances are absent in integrable systems, as a consequence of the (extensive number of) conservation laws associated with integrability.