An improved means of generating high-beta equilibria by injection in magnetostatic simulations of strong ion rings is described. The existence of stochastic orbits in these equilibria is demonstrated. For nonlinear axisymmetric two-dimensional simulations with all three velocity components included, the principal manifestation of such orbits is an eventual violation of left-right mirror symmetry in cases where such symmetry would normally be expected. This effet is due to the exponential divergence of "neighboring" mirror image trajectories. Linearized simulations, in effect, compute the first order separation of orbits which are displaced from each other by an infinitesimal vector for all time. When a linearized code is applied to a problem involving stochastic orbits, the single-particle growth can be faster than that associated with the collective modes of interest, rendering the simulation invalid. This limits the class of problems to which straightforward linearized simulation is applicable. Related difficulties in nonlinear codes using certain "quiet-start" techniques involving loading of particles on axisymmetric rings can be anticipated. These effects should also be evident in simulations of field-reversed mirror systems, ordinary mirror machines, and other devices.