A nonlinear master equation (NLME) is proposed based on general information measures. Classical and cut-off solutions of the NLME are considered. In the former case, the NLME exhibits uniquely defined stationary distributions. In the latter case, there are multiple stationary distributions. In particular, for classical solutions, it is shown that transient solutions converge to stationary distributions that maximize information measures (H-theorem). Cut-off distributions are studied numerically for the Haken-Kelso-Bunz model. The Haken-Kelso-Bunz model is known to describe multistable human motor control systems. It is shown that a stochastic Haken-Kelso-Bunz model based on a NLME can exhibit multiple stationary cut-off distributions. In doing so, we illustrate that multistability in stochastic biological systems can be established by means of cut-off distributions.