  Hybrid hadron-quark equations of state that give rise a third family of
stable compact stars have been shown to be compatible with the LIGO-Virgo event
GW170817. Stable configurations in the third family are called hybrid
hadron-quark stars. The equilibrium stable hybrid hadron-quark star branch is
separated by the stable neutron star branch with a branch of unstable hybrid
hadron-quark stars. The end-state of these unstable configurations has not been
studied, yet, and it could have implications for the formation and existence of
twin stars -- hybrid stars with the same mass as neutron stars but different
radii. We modify existing hybrid hadron-quark equations of state with a
first-order phase transition in order to guarantee a well-posed initial value
problem of the equations of general relativistic hydrodynamics, and study the
dynamics of non-rotating or rotating unstable twin stars via 3-dimensional
simulations in full general relativity. We find that unstable twin stars
naturally migrate toward the hadronic branch. Before settling into the hadronic
regime, these stars undergo (quasi)radial oscillations on a dynamical timescale
while the core bounces between the two phases. Our study suggests that it may
be difficult to form stable twin stars if the phase transition is sustained
over a large jump in energy density, and hence it may be more likely that
astrophysical hybrid hadron-quark stars have masses above the twin star regime.
We also study the minimum-mass instability for hybrid stars, and find that
these configurations do not explode, unlike the minimum-mass instability for
neutron stars. Additionally, our results suggest that oscillations between the
two Quantum Chromodynamic phases could provide gravitational wave signals
associated with such phase transitions in core-collapse supernovae and white
dwarf-neutron star mergers.
