  Non-linear effects in the dynamical evolution of a shearing sheet made of
stars are studied. First the implications of hitherto neglected non-linearities
of the Boltzmann equation for the dynamical evolution of the shearing sheet are
investigated. Using a formalism developed previously on the basis of the
linearized Boltzmann equation it is demonstrated that the inclusion of the
non-linear term leads to a feedback cycle for swing amplified density waves in
the unbounded shearing sheet. Such a feedback is unique to star disks and is
not known for gas disks. In order to present concrete examples of the
non-linear feedback cycle a SCF code was developed and numerical simulations of
the dynamical evolution of the shearing sheet are performed. The numerical
results seem to confirm the theoretical predictions. The evolution of the
shearing sheet resembles closely and might actually explain the recurrent
spiral instabilities found in large-scale numerical simulations of the
dynamical evolution of galactic disks.
