  Non-Hermitian Hamiltonians provide a simple picture for inspecting
dissipative systems with natural or induced gain and loss. We investigate the
Floquet dynamical phase transition in the dissipative periodically time driven
XY and extended XY models, where the imaginary terms represent the physical
gain and loss during the interacting processes with the environment. The
time-independent effective Floquet non-Hermitian Hamiltonians disclose three
regions by analyzing the non-Hermitian gap: pure real gap (real eigenvalues),
pure imaginary gap, and complex gap. We show that each region of the system can
be distinguished by the complex geometrical non-adiabatic phase. We have
discovered that in the presence of dissipation, the Floquet dynamical phase
transitions (FDPTs) still exist in the region where the time-independent
effective Floquet non-Hermitian Hamiltonians reveal real eigenvalues. Opposed
to expectations based on earlier works on quenched systems, our findings show
that the existence of the non-Hermitian topological phase is not an essential
condition for dissipative FDPTs (DFDPTs). We also demonstrate the range of
driven frequency, over which the DFDPTs occur, narrows down by increasing the
dissipation coupling and shrinks to a single point at the critical value of
dissipation. Moreover, quantization and jumps of the dynamical geometric phase
reveals the topological characteristic feature of DFDPTs in the real gap region
where confined to exceptional points.
