  In this paper, we consider local and uniform invariance preserving steplength
thresholds on a set when a discretization method is applied to a linear or
nonlinear dynamical system. For the forward or backward Euler method, the
existence of local and uniform invariance preserving steplength thresholds is
proved when the invariant sets are polyhedra, ellipsoids, or Lorenz cones.
Further, we also quantify the steplength thresholds of the backward Euler
methods on these sets for linear dynamical systems. Finally, we present our
main results on the existence of uniform invariance preserving steplength
threshold of general discretization methods on general convex sets, compact
sets, and proper cones both for linear and nonlinear dynamical systems.
