  In one-dimensional topological superconductors driven periodically with the
frequency $\omega$, two types of topological edge modes may appear, the
well-known Majorana zero mode and a Floquet Majorana mode located at the
quasi-energy $\hbar \omega/2$. We investigate two Josephson-coupled topological
quantum wires in the presence of Coulomb interactions, forming a so-called
Majorana box qubit. An oscillating gate voltage can induce Floquet Majorana
modes in both wires. This allows encoding 3 qubits in a sector with fixed
electron parity. If such a system is prepared by increasing the amplitude of
oscillations adiabatically, it is inherently unstable as interactions
resonantly create quasi particles. This can be avoided by using instead a
protocol where the oscillation frequency is increased slowly. In this case, one
can find a parameter regime where the system remains stable.
