  We investigate semiclassical dynamics of coupled atom-photon interacting
system described by a dimer of anisotropic Dicke model in the presence of
photon loss, exhibiting a rich variety of non-linear dynamics. Based on
symmetries and dynamical classification, we characterize and chart out various
dynamical phases in a phase diagram. A key feature of this system is the
multistability of different dynamical states, particularly the coexistence of
various superradiant phases as well as limit cycles. Remarkably, this dimer
system manifests self-trapping phenomena, resulting in a photon population
imbalance between the cavities. Such a self-trapped state arises from
saddle-node bifurcation, which can be understood from an equivalent
Landau-Ginzburg description. Additionally, we identify a unique class of
oscillatory dynamics self-trapped limit cycle hosting self-trapping of photons.
The absence of stable dynamical phases leads to the onset of chaos, which is
diagnosed using the saturation value of the decorrelator dynamics. Moreover, in
a narrow region, the self-trapped states can coexist with chaotic attractor,
which may have intriguing consequences in quantum dynamics. Finally, we discuss
the experimental relevance of our findings, which can be tested in cavity and
circuit quantum electrodynamics setups.
