Predicting instabilities of a tunable ring laser with an iterative map model

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