  Hermitian Hamiltonians with time-periodic coefficients can be analyzed via
Floquet theory, and have been extensively used for engineering Floquet
Hamiltonians in standard quantum simulators. Generalized to non-Hermitian
Hamiltonians, time-periodicity offers avenues to engineer the landscape of
Floquet quasi-energies across the complex plane. We investigate two-level
non-Hermitian Hamiltonians with coefficients that have different periodicities
using Floquet theory. By analytical and numerical calculations, we obtain their
regions of stability, defined by real Floquet quasi-energies, and contours of
exceptional point (EP) degeneracies. We extend our analysis to study the phases
that accompany the cyclic changes. Our results demonstrate that time-periodic,
non-Hermitian Hamiltonians generate a rich landscape of stable and unstable
regions.
