The true motion of a particle in a one-dimensional potential well is regular, since conservation of energy constrains the velocity ν each value of the coordinate x. Nonetheless, when the orbit is computed numerically, stochastic behavior can result. We have considered simple integrators as mappings from ( x, ν) at one discrete time level to ( x, ν) at the next. In general, when the timestep size Δ is small enough there are closed orbits, while for larger values there is chaos. Chaos can result for surprisingly small values of Δ in cases where the physical phase plane includes a separatrix. The behavior of the leapfrog mover as applied to motion in a particular double-well potential is examined in detail. Here, the onset of Stochasticity occurs at step sizes much smaller than the stability threshold associated with the harmonic dependence of the potential at large | x|. Other one-dimensional wells and movers are also treated; implications of the area-preserving and energy conserving attributes possessed by some movers are discussed. A new variant of the Standard Map, displaying symmetry about both x = 0 and ν = 0 in its phase plane, is introduced.