  Dynamical quantum phase transitions (DQPTs) are characterized by nonanalytic
behaviors of physical observables as functions of time. When a system is
subject to time-periodic modulations, the nonanalytic signatures of its
observables could recur periodically in time, leading to the phenomena of
Floquet DQPTs. In this work, we systematically explore Floquet DQPTs in a class
of periodically quenched one-dimensional system with chiral symmetry. By tuning
the strength of quench, we find multiple Floquet DQPTs within a single driving
period, with more DQPTs being observed when the system is initialized in
Floquet states with larger topological invariants. Each Floquet DQPT is further
accompanied by the quantized jump of a dynamical topological order parameter,
whose values remain quantized in time if the underlying Floquet system is
prepared in a gapped topological phase. The theory is demonstrated in a
piecewise quenched lattice model, which possesses rich Floquet topological
phases and is readily realizable in quantum simulators like the
nitrogen-vacancy center in diamonds. Our discoveries thus open a new
perspective for the Floquet engineering of DQPTs and the dynamical detection of
topological phase transitions in Floquet systems.
