We investigate the rich nonequilibrium physics arising in periodically driven open quantum systems, specifically those realized in microcavity resonators, whose dynamics are governed by a non-Hermitian Hamiltonian hosting Floquet exceptional points (FEPs). By introducing a periodically quenched driving protocol, we analytically derive the Floquet effective Hamiltonian and determine the locations of multiple FEPs harbored within the Floquet bulk bands. We demonstrate that the pair production and annihilation of these FEPs can be precisely controlled by fine-tuning the system parameters, and zero and $\pi$ FEPs are topologically characterized by a robust integer-quantized winding numbers. To probe these singularities, we introduce a biorthogonal Floquet fidelity susceptibility whose value exhibits large nonzero peaks at the momentum points hosting FEPs in the Brillouin zone. Furthermore, the momentum-summed susceptibility displays a sharp divergence when the number of FEPs change with respect to the time period of the drive. Our findings also reveal the emergence of Floquet edge states around zero energy and Dirac-like dispersion around $\pi$. Moreover, our model reveals a higher-order skin effect, where the periodically driven Hamiltonian hosts skin modes localized at both edges and corners. These insights offer novel avenues for the Floquet engineering of topological singularities in driven dissipative systems, with significant potential for manipulating light and matter at the microscale.

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• Edge states
• Skin effect
• Floquet systems
• Non-Hermitian systems
• Topological materials
• Tight-binding model