We study a non-Hermitian chiral topological superconductor system on two dimensional square lattice, from which we obtained a rich topological phase diagram and established an exact relationship between topological charge flow of exceptional points in generalized Brillouin zone and change of topological properties. Its rich topological phase diagram is the result of competition between anisotropy and non-Hermitian effect. This system belongs to class D according to AZ classification of non-Hermitian systems. Each topological phase can be characterized by a 2D Z number, which indicates the number of chiral edge modes, and two 1D Z2numbers, which indicate the existence of zero modes at edge dislocations.

Floquet engineering plays a key role in realizing novel dynamical topological states. The conventional Floquet engineering, however, only applies to time-periodic non-dissipative Hermitian systems, and for the open quantum systems, non-Hermitian processes usually occur. So far, it remains unclear how to characterize the topological phases of time-periodic open quantum systems via the frequency space Floquet Hamiltonian. Here, we propose the non-Floquet theory to solve the problem and illustrate it by a continuously time-periodic non-Hermitian bipartite chain. In non-Floquet theory, a temporal non-unitary transformation is exercised on the Floquet states, and the transformed Floquet spectrum restores the form of the Wannier-Stark ladder. Besides, we also show that different choices of the starting points of the driving period can result in different localization behavior, effects of which can reversely be utilized to design quantum detectors of phases in dissipative oscillating fields. Our methods are capable of describing topological features in dynamical open quantum systems with various driving types and can find its applications to construct new types of dynamical topological materials.