  Outbursting AM CVn stars exhibit outbursts similar to those observed in
different types of dwarf novae. Their light-curves combine the characteristic
features of SU UMa, ER UMa, Z Cam, and WZ Sge-type systems but also show a
variety of properties never observed in dwarf novae. The compactness of AM CVn
orbits and their unusual chemical composition make these systems valuable
testbeds for outburst models. We aim for a better understanding of the role of
helium in the accretion disc instability mechanism, testing the model for dwarf
novae outbursts in the case of AM CVn stars, and aim to explain the outburst
light-curves of these ultra-compact binaries. We calculated the properties of
the hydrogen-free AM CVn stars using our previously developed numerical code
adapted to the different chemical composition of these systems and supplemented
with formulae accounting for mass transfer rate variations, additional sources
of the disc heating, and the primary's magnetic field. We discovered how
helium-dominated discs react to the thermal-viscous instability and were able
to reproduce various features of the outburst cycles in the light-curves of AM
CVn stars. The AM CVn outbursts can be explained by the suitably adapted
dwarf-nova disc instability model but, as in the case of its application to
hydrogen-dominated cataclysmic variables, one has to resort to additional
mechanisms to account for the observed superoutbursts, dips, cycling states,
and standstills. We show that the enhanced mass-transfer rate, due presumably
to variable irradiation of the secondary, must not only be taken into account
but is a determining factor that shapes AM CVn star outbursts. The cause of the
variable secondary's irradiation has yet to be understood; the best candidate
is the precession of a tilted/warped disc.
