A powerful extension of our research made possible by the cold-atom context is a Floquet generalization of anyon physics. Time-translation symmetry can be explicitly and controllably broken by strong periodic driving, allowing for full time-dependent control over all elements of the Kitaev chain Hamiltonian. In the Floquet realization of the Kitaev chain, if the two terms giving rise to trivial and nontrivial phases are applied sequentially and periodically with a varying duty cycle, it is possible to map out a topological Floquet phase diagram in which the signatures of Majorana-supporting phases emerge as a straightforwardly measurable nonlinear response of the system to the periodic drive. Remarkably, new anyonic phases distinguished by their temporal periodicity are predicted to appear as so-called "Floquet time crystals." This would realize not only a new species of anyons but also a new class of experimental anyon detection and control abilities.