  Mode-locked lasers play the role of the ideal testbeds for studying
self-coherent structures, dissipative solitons, with stable spatiotemporal
profiles supported by the balance between dispersion and nonlinearity. However,
under some conditions, the profile and energy of the solitons can oscillate
(breath) periodically. Unlike the previous studies of different scalar
mechanisms of breathers emergence, we demonstrate experimentally and
theoretically a new vector mechanism where breathing of the output power is a
result of the heteroclinic dynamics. The heteroclinic orbit is a trajectory
periodically evolving nearby the neighborhood of one of the orthogonal states
of polarization with further switching to and evolving nearby the other state
of polarization. The dwelling time for the trajectory near each orthogonal
state is determined by the cavity anisotropy adjustable with the help of the
polarization controller for the pump wave. The obtained results on tunability
of breathing dynamics enables unlocking novel techniques of the flexible
control and manipulation of the polarization dynamics of dissipative solitons.
