Three-dimensional (3D) hydrodynamic simulations of shell oxygen burning exhibit bursty, recurrent fluctuations in turbulent kinetic energy. These are shown to be due to a general instability of the convective cell, requiring only a localized source of heating or cooling. Such fluctuations are shown to be suppressed in simulations of stellar evolution which use the mixing-length theory. Quantitatively similar behavior occurs in the model of a convective roll (cell) of Lorenz, which is known to have a strange attractor that gives rise to chaotic fluctuations in time of velocity and, as we show, luminosity. Study of simulations suggests that the behavior of a Lorenz convective roll may resemble that of a cell in convective flow. We examine some implications of this simplest approximation and suggest paths for improvement. Using the Lorenz model as representative of a convective cell, a multiple-cell model of a convective layer gives total luminosity fluctuations which are suggestive of irregular variables (red giants and supergiants), and of the long secondary period feature in semiregular asymptotic giant branch variables. This "τ-mechanism" is a new source for stellar variability, which is inherently nonlinear (unseen in linear stability analysis), and one closely related to intermittency in turbulence. It was already implicit in the 3D global simulations of Woodward et al. This fluctuating behavior is seen in extended two-dimensional simulations of CNeOSi burning shells, and may cause instability which leads to eruptions in progenitors of core-collapse supernovae prior to collapse.