  We study the classical and quantum theory of a class of nonlinear
differential equations on chronology violating spacetime models in which space
consists of finitely many discrete points. Classically, in the linear and
weakly nonlinear regimes (for generic choices of parameters) we prove existence
and uniqueness of solutions corresponding to initial data specified before the
dischronal region; however, uniqueness (but not existence) fails in the
strongly coupled regime. The evolution preserves the symplectic structure.
  The quantum theory is approached via the quantum initial value problem
(QIVP); that is, by seeking operator-valued solutions to the equation of motion
with initial data representing the canonical (anti)commutation relations. Using
normal operator ordering, we construct solutions to the QIVP for both Bose and
Fermi statistics (again for generic choice of parameters) and prove that these
solutions are unique. For models with two spatial points, the resulting
evolution is unitary; however, for a more general model the evolution fails to
preserve the (anti)commutation relations and is therefore nonunitary. We show
that this nonunitary evolution cannot be described using a superscattering
operator with the usual properties.
  We present numerical evidence to show that the bosonic quantum theory can
pick out a unique classical limit for certain ranges of the coupling strength,
even when there are many classical solutions. We show that the quantum theory
depends strongly on the choice of operator ordering.
  In addition, we show that our results differ from those obtained using the
``self-consistent path integral''. It follows that the path integral evolution
does not correspond to a solution of the equation of motion.
