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Based on a rate equation description of a directly modulated laser diode, we investigate the origin of intermittent bursts that appear in the output power of the device. Sudden drops in the maximum amplitude of the laser's output characterize this behavior, found in the period doubling sequence that appears when the modulation index varies. From the investigation of this behavior the paper reveals the cause for the extreme sensitivity to noise of modulated laser diodes. Using bifurcation theory techniques, we explain how the unstable periodic solutions and the intrinsic noise of the device play a key role in the mechanism giving rise to these bursts. The paper shows that the mechanism is composed of two fundamental stages, which have been referred to as capture and reinjection. As we clarify the origin of the two stages, the different time scales involved in each one are also identified. From this analysis, we can conclude that the intermittent bursts are a characteristic signature of the process by which the laser stochastic dynamics progressively embeds one of the saddle cycles as the modulation index is varied.