  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.
