Hysteresis of dissipative soliton molecules in mode-locked fiber lasers.

We numerically demonstrate bistability of a robust dissipative soliton molecule with independently evolving phase in a normal dispersion mode-locked fiber laser. It is shown that the separation of the peaks exhibits a hysteretic behavior as a function of the pump energy. Physically this corresponds to the coexistence of two bound-state branches, one with chaotic dynamics and another one with periodic dynamics. The crucial role of gain saturation for the appearance of the hysteresis is revealed. We observe the period-doubling route to chaos.