  We introduce a Floquet spinor Bose-Einstein condensate induced by a
periodically driven quadratic Zeeman coupling whose frequency is larger than
any other energy scales. By examining a spin-1 system available in ultracold
atomic gases, we demonstrate that such an external driving field has great
effect on the condensate through emergence of a unique spin-exchange
interaction. We uncover that the ferromagnetic condensate has several
unconventional stationary states and thus exhibits rich continuous phase
transitions. On the other hand, the antiferromagnetic condensate is found to
possess a nontrivial metastable region, which supports unusual elementary
excitations and hysteresis phenomena.
