  Simple mathematical models have been unable to predict the conditions leading
to instabilities in a tuneable ring laser. Here, we propose a nonlinear
iterative map model for tuneable ring lasers. Solving a reduced nonlinear
Schr\"odinger equation for each component in the laser cavity, we obtain an
algebraic map for each component. Iterating through the maps gives the total
effect of one round trip. By neglecting the nonlinearity, we find a linearly
chirped Gaussian to be the analytic fixed point solution, which we analyze
asymptotically. We then numerically solve the full nonlinear model, allowing us
to probe the underlying interplay of dispersion, modulation, and nonlinearity
as the pulse evolves over hundreds of round trips of the cavity. In the
nonlinear case, we find the chirp saturates, and the Fourier transform of the
pulse becomes more rectangular in shape. Finally, for a nominal plane in the
parameter space, we uncover a rich, sharp boundary separating the stable region
and the unstable region where modulation instability degrades the pulse into an
unsustainable state.
