Bifurcation diagram of a complex delay-differential equation with cubic nonlinearity

We reduce the Lang-Kobayashi equations for a semiconductor laser with external optical feedback to a single complex delay-differential equation in the long delay-time limit. The reduced equation has a time-delayed linear term and a cubic instantaneous nonlinearity. There are only two parameters, the real linewidth enhancement factor and the complex feedback strength. The equation displays a very rich dynamics and can sustain steady, periodic, quasiperiodic, and chaotic regimes. We study the steady solutions analytically and analyze the periodic solutions by using a numerical continuation method. This leads to a bifurcation diagram of the steady and periodic solutions, stable and unstable. We illustrate the chaotic regimes by a direct numerical integration and show that low frequency fluctuations still occur.