  We show how transitions between different Lifshitz phases in bilayer Dirac
materials with and without spin-orbit coupling can be studied by driving the
system. The periodic driving is induced by a laser and the resultant phase
diagram is studied in the high frequency limit using the Brillouin-Wigner
perturbation approach to leading order. The examples of such materials include
bilayer graphene and spin-orbit coupled materials such as bilayer silicene. The
phase diagrams of the effective static models are analyzed to understand the
interplay of topological phase transitions, with changes in the Chern number
and topological Lifshitz transitions, with the ensuing changes in the Fermi
surface. Both the topological transitions and the Lifshitz transitions are
tuned by the amplitude of the drive.
