Period tripling and chaos in the dynamic behavior of directlymodulated diode lasers

The paper presents the relevance of period-tripling behavior that has recently been found in different experimental studies of directly modulated laser diodes. Applying different numerical techniques to the rate equation model of the laser diode, among which we highlight the continuation method to calculate the unstable solutions of the system, we show that period-tripling behavior appears and disappears in two tangent bifurcations. Therefore, the period-three solutions form a closed bifurcation curve called isola. In between these two tangent bifurcations, the period-three solution coexists with the chaotic attractor reached by a period-doubling cascade, giving rise to a hysteresis loop in the deterministic case. Also, we have found that a boundary crisis might be behind the chaotic behavior that is observed for the highest values of the modulation index. The effects of random noise fluctuations in the laser diode dynamics are also studied. Langevin noise sources are included in the rate equation model and appropriate stochastic integration methods have been used. The route to chaos that we have obtained points out the relevant role that noise has in achieving agreement between numerical studies and experimental results that have been published. The introduction of noise has been proved to be of major importance in determining the system behavior in the regions of the coexistence of solutions