  We consider a U(1)-invariant nonlinear Dirac equation in dimension $n=3$,
interacting with itself via the mean field mechanism. We analyze the long-time
asymptotics of solutions and prove that, under certain generic assumptions,
each finite charge solution converges as $t\to\pm\infty$ to the two-dimensional
set of all "nonlinear eigenfunctions" of the form $\phi(x)e\sp{-i\omega t}$.
This global attraction is caused by the nonlinear energy transfer from lower
harmonics to the continuous spectrum and subsequent dispersive radiation.
  The research is inspired by Bohr's postulate on quantum transitions and
Schr\"odinger's identification of the quantum stationary states to the
nonlinear eigenfunctions of the coupled U(1)-invariant Maxwell-Schr\"odinger
and Maxwell-Dirac equations.
