  We study Floquet eigenspectra of a nonlinear two-mode system under a periodic
driving of the off-diagonal coupling. By solving the Gross-Pitaevskii equation
numerically, we obtain triangular and loop structures near the crossings of
different Floquet branches. At lower driving frequencies, we find "ring" and
"double-ring" structures which are distinct from the well-known loop structure.
The mechanism of the emergence of these structures is discussed and the
parameter windows of their existence are obtained analytically. In addition, we
study the evolution of the system under the driving with an adiabatic sweep and
find there are some dynamically unstable states in the Floquet eigenspectra
which break the quantum adiabaticity.
