  Boundary time crystals (BTC's) are non-equilibrium phases of matter occurring
in quantum systems in contact to an environment, for which a macroscopic
fraction of the many body system breaks time translation symmetry. We study
BTC's in collective $d$-level systems, focusing in the cases with $d=2$, $3$
and $4$. We find that BTC's appear in different forms for the different cases.
We first consider the model with collective $d=2$-level systems [presented in
Phys. Rev. Lett. $121$, $035301$ ($2018$)], whose dynamics is described by a
Lindblad master equation, and perform a throughout analysis of its phase
diagram and Jacobian stability for different interacting terms in the coherent
Hamiltonian. In particular, using perturbation theory for general (non
Hermitian) matrices we obtain analytically how a specific $\mathbb{Z}_2$
symmetry breaking Hamiltonian term destroys the BTC phase in the model. Based
on these results we define a $d=4$ model composed of a pair of collective
$2$-level systems interacting with each other. We show that this model support
richer dynamical phases, ranging from limit-cycles, period-doubling
bifurcations and a route to chaotic dynamics. The BTC phase is more robust in
this case, not annihilated by the former symmetry breaking Hamiltonian terms.
The model with collective $d=3$-level systems is defined similarly, as
competing pairs of levels, but sharing a common collective level. The dynamics
can deviate significantly from the previous cases, supporting phases with the
coexistence of multiple limit-cycles, closed orbits and a full degeneracy of
zero Lyapunov exponents.
