  We study the entanglement properties in non-equilibrium quantum systems with
the SU(1, 1) structure. Through M\"obius transformation, we map the dynamics of
these systems following a sudden quench or a periodic drive onto three distinct
trajectories on the Poincar\'e disc, corresponding the heating, non-heating,
and a phase boundary describing these non-equilibrium quantum states. We
consider two experimentally feasible systems where their quantum dynamics
exhibit the SU(1, 1) structure: the quench dynamics of the Bose-Einstein
condensates and the periodically driven coupled oscillators. In both cases, the
heating, non-heating phases, and their boundary manifest through distinct
signatures in the phonon population where exponential, oscillatory, and linear
growths classify these phases. Similarly, the entanglement entropy and
negativity also exhibit distinct behaviors (linearly, oscillatory, and
logarithmic growths) characterizing these phases, respectively. Notibly, for
the periodically driven coupled oscillators, the non-equilibrium properties are
characterized by two sets of SU(1, 1) generators. The corresponding two sets of
the trajectories on two Poincar\'e discs lead to a more complex phase diagram.
We identify two distinct phases within the heating region discernible solely by
the growth rate of the entanglement entropy, where a discontinuity is observed
when varying the parameters across the phase boundary within in heating region.
This discontinuity is not observed in the phonon population.
