Dispersive Time-Delay Dynamical Systems

We present a theoretical approach to investigate the effect of dispersion in dynamical systems commonly described by time-delay models. The introduction of a polarization equation provides a means to introduce dispersion as a distributed delay term. The expansion of this term in power series, as usually performed to study the propagation of waves in spatially extended systems, can lead to the appearance of spurious instabilities. This approach is illustrated using a long cavity laser, where in the normal dispersion regime both the experiment and theory show a stable operation, while a modulation instability, commonly referred as the Benjamin-Feir instability, is observed in the anomalous dispersion regime.