  Creating crystal bilayers twisted with respect to each other would lead to
large periodic supercell structures, which can support a wide range of novel
electron correlated phenomena, where the full understanding is still under
debate. Here, we propose a new scheme to realize a nonlinearity-induced
dynamical self-organized twisted-bilayer lattice in an atomic Bose-Einstein
condensate (BEC). The key idea here is to utilize the nonlinear effect from the
intrinsic atomic interactions to couple different layers and induce a dynamical
self-organized supercell structure, dramatically distinct from the conventional
wisdom to achieve the static twisted-bilayer lattices. To illustrate that, we
study the dynamics of a two-component BEC and show that the nonlinear
interaction effect naturally emerged in the Gross-Pitaevskii equation of
interacting bosonic ultracold atoms can dynamically induce both periodic
(commensurable) and aperiodic (incommensurable) moir\'{e} structures. One of
the interesting moir\'{e} phenomena, i.e., the flat-band physics, is shown
through investigating the dynamics of the wave packet of BEC. Our proposal can
be implemented using available state-of-the-art experimental techniques and
reveal a profound connection between the nonlinearity and twistronics in cold
atom quantum simulators.
