Studies of stochastic or ergodic behavior of beam particle orbits in axisymmetric systems with field reversal produced by ion rings or by neutral injection are presented. In the former case a large class of orbits is ergodic, whereas in the latter most are integrable. Effects of ergodic behavior on particle confinement, equilibrium, magnetic compression, and stability are discussed. The modification, due to ergodic orbits, of the stability criterion for low frequency ( omega << omega _ci) resonant instabilities is presented.