  We investigate the Gross-Pitaevskii equation for a classically chaotic
system, which describes an atomic Bose-Einstein condensate confined in an
optical lattice and driven by a spatiotemporal periodic laser field. It is
demonstrated that the exact Floquet states appear when the external
time-dependent potential is balanced by the nonlinear mean-field interaction.
The balance region of parameters is divided into a phase-continuing region and
a phase-jumping one. In the latter region, the Floquet states are
spatiotemporal vortices of nontrivial phase structures and zero-density cores.
Due to the velocity singularities of vortex cores and the blowing-up of
perturbed solutions, the spatiotemporal vortices are unstable periodic states
embedded in chaos. The stability and instability of these Floquet states are
numerically explored by the time evolution of fidelity between the exact and
numerical solutions. It is numerically illustrated that the stable Floquet
states could be prepared from the uniformly initial states by slow growth of
the external potential.
