  Gauge fields are a central concept in fundamental theories of physics, and
responsible for mediating long-range interactions between elementary particles.
Recently, it has been proposed that dynamical gauge fields can be naturally
engineered by photons in composite, neutral quantum gas--cavity systems using
suitable atom-photon interactions. Here we comprehensively investigate
nonequilibrium dynamical phases appearing in a two-leg bosonic lattice model
with leg-dependent, dynamical complex tunnelings mediated by cavity-assisted
two-photon Raman processes. The system constitutes a minimal dynamical
flux-lattice model. We study fixed points of the equations of motion and their
stability, the resultant dynamical phase diagram, and the corresponding phase
transitions and bifurcations. Notably, the phase diagram features a plethora of
nonequilibrium dynamical phases including limit-cycle and chaotic phases. In
the end, we relate regular periodic dynamics (i.e., limit-cycle phases) of the
system to time crystals.
