  We study local and non-local Polyakov Nambu-Jona-Lasinio models and analyze
their respective phase transition diagram. We construct hybrid stars using the
zero temperature limit of the local and non-local versions of
Nambu-Jona-Lasinio model for quark matter and the modern GM1(L) parametrization
of the non-linear relativistic mean field model for hadronic matter. We compare
our models with data from PSR J1614-2230 and PSR J0343+0432 and also from
GW170817 and its electromagnetic counterpart GRB170817A and AT2017gfo. We study
observational signatures of the appearance of a mixed phase as a result of
modeling a phase transition that mimics the Gibbs formalism and compare the
results with the sharp first-order phase transition obtained using the Maxwell
construction. We also study in detail the $g$-mode associated with
discontinuities in the equation of state, and calculate non-radial oscillation
modes using relativistic Cowling approximation.
