  We show that even small dissipation can strongly affect the
Fermi-Pasta-Ulam-Tsingou recurrence phenomenon. Taking the fully nonlinear
stage of modulational instability as an experimentally accessible example, we
show that the linear attenuation induces the symmetry of the recurrence to be
broken through separatrix crossing occurring at multiple critical values of the
attenuation. We provide experimental evidence for this phenomenon in a fiber
optics experiment designed in such a way that the effective losses can be
carefully tailored by techniques based on Raman amplification.
