  We investigate the dynamical characterization theory for periodically driven
systems in which Floquet topology can be fully detected by emergent topological
patterns of quench dynamics in momentum subspaces called band-inversion
surfaces. We improve the results of a recent work [Zhang et al., Phys. Rev.
Lett. 125, 183001 (2020)] and propose a more flexible scheme to characterize a
generic class of $d$-dimensional Floquet topological phases classified by
$\mathbb{Z}$-valued invariants by applying a quench along an arbitrary
spin-polarization axis. Our basic idea is that by disassembling the Floquet
system into multiple static subsystems that are periodic in quasienergy, a full
characterization of Floquet topological phases reduces to identifying a series
of bulk topological invariants for time-independent Hamiltonians, which greatly
enhances the convenience and flexibility of the measurement. We illustrate the
scheme by numerically analyzing two experimentally realizable models in two and
three dimensions, respectively, and adopting two different but equivalent
viewpoints to examine the dynamical characterization. Finally, considering the
imperfection of experiment, we demonstrate that the present scheme can also be
applied to a general situation where the initial state is not completely
polarized. This study provides an immediately implementable approach for
dynamically classifying Floquet topological phases in ultracold atoms or other
quantum simulators.
