  Having the potential for performing quantum computation, topological
superconductors have been generalized to the second-order case. The
hybridization of different orders of topological superconductors is attractive
because it facilitates the simultaneous utilization of their respective
advantages. However, previous studies found that they cannot coexist in one
system due to the constraint of symmetry. We propose a Floquet engineering
scheme to generate two-dimensional (2D) hybrid-order topological
superconductors in an odd-parity superconductor system. Exotic hybrid-order
phases exhibiting the coexisting gapless chiral edge states and gapped Majorana
corner states not only in two different quasienergy gaps but also in one single
quasienergy gap are created by periodic driving. The generalization of this
scheme to the 3D system allows us to discover a second-order Dirac
superconductor featuring coexisting surface and hinge Majorana Fermi arcs. Our
results break a fundamental limitation on topological phases and open a
feasible avenue to realize topological superconductor phases without static
analogs.
