The comprehensive parameter space of self-oscillation and its period-doubling route to chaos are shown for bounded beam-plasma systems. In this parametrization, it is helpful to use a potentially universal parameter in close analogy with free-electron-laser chaos. A common parameter, which is related to the velocity slippage and the ratio of bounce to oscillation frequencies, is shown to have similar significance for different physical systems. This single parameter replaces the dependences on many input parameters, thus suitable for a simplifying and diagnostic measure of nonlinear dynamical and chaotic phenomena for various systems of particle-wave interactions. The results of independent kinetic simulations verify those of nonlinear fluid simulations. ©1998 American Institute of Physics.