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The long-time asymptotics is analyzed for all finite energy solutions to amodel $\mathbf{U}(1)$-invariant nonlinear Dirac equation in one dimension,coupled to a nonlinear oscillator: {\it each finite energy solution} convergesas $t\to\pm\infty$ to the set of all `nonlinear eigenfunctions' of the form$(\psi_1(x)e^{-i\omega_1 t},\psi_2(x)e^{-i\omega_2 t})$. The {\it globalattraction} is caused by the nonlinear energy transfer from lower harmonics tothe continuous spectrum and subsequent dispersive radiation.  We justify this mechanism by the strategy based on \emph{inflation ofspectrum by the nonlinearity}. We show that any {\it omega-limit trajectory}has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the originalequation. This equation implies the key {\it spectral inclusion} for spectrumof the nonlinear term. Then the application of the Titchmarsh convolutiontheorem reduces the spectrum of $j$-th component of the omega-limit trajectoryto a single harmonic $\omega_j\in[-m,m]$, $j=1,2$.

Global attractor for 1D Dirac field coupled to nonlinear oscillator