  We investigated experimentally non-equilibrium state of a two-dimensional
electron gas (2DEG) in the quantum Hall effect (QHE) regime, studying the
hysteresis of magnetoresistance of a 2DEG with a constriction. The large
amplitude of the hysteresis enabled us to make the consistent phenomenological
description of the hysteresis. We studied the dependence on the magnetic field
sweep prehistory (minor loop measurements), recovered the anhysteretic curve,
and studied the time dependence of the magnetoresistance. We showed that the
hysteresis of magnetoresistance of a 2DEG in the QHE regime has significant
phenomenological similarities with the hysteresis of magnetization of
ferromagnetic materials, showing multistability, jumps of relaxation, and
having the anhysteretic curve. Nevertheless, we revealed the crucial
difference, manifested itself in an unusual inverted (anti-coercive) behavior
of the magnetoresistance hysteresis. The time relaxation of the hysteresis has
fast and slow regimes, similar to that of non-equilibrium magnetization of a
2DEG in QHE regime pointing to their common origin. We studied the dependence
of the hysteresis loop area on the lithographic width of the constriction and
found the threshold value of width $\sim$1.35 $\mu$m beyond which the
hysteresis is not observed. This points to the edge nature of the
non-equilibrium currents (NECs) and allows us to determine the width of the
NECs area ($\sim$0.5 $\mu$m). We suggest the qualitative picture of the
observed hysteresis, based on non-equilibrium redistribution of the electrons
among the Landau level states and assuming huge imbalance between the
population of bulk and edge electronic states.
