A two-dimensional flow model is introduced with deterministic behavior consisting of bursts that become successively larger, with longer interburst time intervals between them. The system is symmetric in one variable and there are bursts on either side of , separated by the presence of an invariant manifold at . In the presence of arbitrarily small additive noise in the direction, the successive bursts have bounded amplitudes and interburst intervals. This system with noise is proposed as a model for edge-localized modes in tokamaks. With noise, the bursts can switch from positive to negative and vice versa. The probability distribution of burst heights and interburst periods is studied, as is the dependence of the statistics on the noise variance. The modification of this behavior as the symmetry in is broken is studied, showing qualitatively similar behavior if the symmetry breaking is small enough. Experimental observations of a nonlinear circuit governed by the same equations are presented, showing good agreement.