  Ideally rigid objects establish sustained contact with one another via
complete chatter (a.k.a. Zeno behavior), i.e. an infinite sequence of
collisions accumulating in finite time. Alternatively, such systems may also
exhibit a finite sequence of collisions followed by separation (sometimes
called incomplete chatter). Earlier works concerning the chattering of slender
rods in two dimensions determined the exact range of model parameters, where
complete chatter is possible. We revisit and slightly extend these results.
Then the bulk of the paper examines the chattering of three-dimensional objects
with multiple points hitting an immobile plane almost simultaneously. In
contrast to rods, the motion of these systems is complex, nonlinear, and
sensitive to initial conditions and model parameters due to the possibility of
various impact sequences. These difficulties explain why we model this
phenomenon as a nondeterministic discrete dynamical system. We simplify the
analysis by assuming linearized kinematics, frictionless interaction, by
neglecting the effect of external forces, and by investigating objects with
rotational symmetry. Application and extension of the theory of common
invariant cones of multiple linear operators enable us to find sufficient
conditions of the existence of initial conditions, which give rise to complete
chatter. Additional analytical and numerical investigations predict that our
sufficient conditions are indeed exact, moreover solving a simple eigenvalue
problem appears to be enough to judge the possibility of complete chatter.
\keywords{contact dynamics \and chattering \and Zeno behavior \and common
invariant cone
