  We study the non-axisymmetric evolution of magnetized clouds, using a 2D MHD
code based on the physically motivated thin-disk approximation. We found that
such clouds become unstable to non-axisymmetric perturbations after the
supercritical cores are formed due to ambipolar diffusion. We show that for a
wide range of initial cloud parameters, the $m=2$ mode grows nonlinearly into a
bar during the isothermal collapse after the supercritical core formation. The
supercritical core can break up into fragments during or after the isothermal
phase of cloud evolution. The outcome of fragmentation depends on the initial
cloud conditions, such as the magnetic field strength, rotation rate, amount of
cloud mass and mass distribution. It is classified into three different types:
(1) separate core formation, (2) bar fragmentation, and (3) disk fragmentation.
These three types of fragmentation loosely correspond to the empirical
classification of embedded binary and multiple systems of Looney, Mundy, &
Welch, based on millimeter dust continuum observations. The well-studied
starless core, L1544, appears to belong to the bar fragmentation type. We
expect it to produce a highly elongated, opaque bar at the center in the
future, which should break up into fragments of initial masses in the
substellar regime.
