  Open systems with gain, loss, or both, described by non-Hermitian
Hamiltonians, have been a research frontier for the past decade. In particular,
such Hamiltonians which possess parity-time ($\mathcal{PT}$) symmetry feature
dynamically stable regimes of unbroken symmetry with completely real
eigenspectra that are rendered into complex conjugate pairs as the strength of
the non-Hermiticity increases. By subjecting a $\mathcal{PT}$-symmetric system
to a periodic (Floquet) driving, the regime of dynamical stability can be
dramatically affected, leading to a frequency-dependent threshold for the
$\mathcal{PT}$-symmetry breaking transition. We present a simple model of a
time-dependent $\mathcal{PT}$-symmetric Hamiltonian which smoothly connects the
static case, a $\mathcal{PT}$-symmetric Floquet case, and a
neutral-$\mathcal{PT}$-symmetric case. We analytically and numerically analyze
the $\mathcal{PT}$ phase diagrams in each case, and show that slivers of
$\mathcal{PT}$-broken ($\mathcal{PT}$-symmetric) phase extend deep into the
nominally low (high) non-Hermiticity region.
