  We report the theoretical discovery and characterization of higher-order
Floquet topological phases dynamically generated in a periodically driven
system with mirror symmetries. We demonstrate numerically and analytically that
these phases support lower-dimensional Floquet bound states, such as corner
Floquet bound states at the intersection of edges of a two-dimensional system,
protected by the nonequilibrium higher-order topology induced by the periodic
drive. We characterize higher-order Floquet topologies of the bulk Floquet
Hamiltonian using mirror-graded Floquet topological invariants. This allows for
the characterization of a new class of higher-order "anomalous" Floquet
topological phase, where the corners of the open system host Floquet bound
states with the same as well as with double the period of the drive. Moreover,
we show that bulk vortex structures can be dynamically generated by a drive
that is spatially inhomogeneous. We show these bulk vortices can host multiple
Floquet bound states. This "stirring drive protocol" leverages a connection
between higher-order topologies and previously studied fractionally charged,
bulk topological defects. Our work establishes Floquet engineering of
higher-order topological phases and bulk defects beyond equilibrium
classification and offers a versatile tool for dynamical generation and control
of topologically protected Floquet corner and bulk bound states.
