  Hot Jupiters, orbiting their host stars at extremely close distances, undergo
tidal evolution, with some being engulfed by their stars due to angular
momentum exchanges induced by tidal forces. However, achieving double
synchronization can prolong their survival. Using the MESA stellar evolution
code, combined with the magnetic braking model of Matt et al. (2015), we
calculate 25,000 models with different metallicity and study how to attain the
conditions that trigger the long-term double synchronization. Our results
indicate that massive planets orbiting stars with lower convective turnover
time are easier to achieve long-term double synchronization. The rotation
angular velocity at the equilibrium point ($\Omega_{\mathrm{sta}}$) is almost
equal to orbital angular velocity of planet ($\mathrm{n}$) for the majority of
the main sequence lifetime if a system has undergone a long-term double
synchronization, regardless of their state at this moment. We further compared
our results with known parameters of giant planetary systems and found that
those systems with larger planetary masses and lower convective turnover time
seem to be less sensitive to changes in the tidal quality factor $Q'_{_*}$. We
suggest that for systems that fall on the state of $\Omega_{\mathrm{sta}}
\approx n$, regardless of their current state, the synchronization will persist
for a long time if orbital synchronization occurs at any stage of their
evolution. Our results can be applied to estimate whether a system has
experienced long-term double synchronization in the past or may experience it
in the future.
