  A honeycomb array of helical waveguides with zigzag-zigzag edges and a
refractive index gradient orthogonal to the edges may support Floquet bound
states in continuum (BICs). The gradient of the refractive index leads to
strong asymmetry of the Floquet-Bloch spectrum. The mechanism of creation of
such Floquet BICs is understood as emergence of crossings and avoided crossings
of the branches supported by spatially limited stripe array. The whole spectrum
of a finite array is split into the bulk branches being continuation of the
edge states in the extended zone revealing multiple self-crossings and bulk
modes disconnected from the gap states by avoided crossing. Nearly all states
in the system are localized due to the gradient, but topological edge states
manifest much stronger localization than other states. Such strongly localized
Floquet BICs coexist with localized Wannier-Stark-like bulk modes. Robustness
of the edge Floquet states is confirmed by their passage through a localized
edge defect in the form of a missing waveguide.
