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.
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