Synchronization, fractional entrainment and chaotic hysteresis of driven breathing soliton molecules in ultrafast lasers

Breathing soliton molecules exhibit unique internal dynamics that map onto a wide range of physical systems, providing a versatile platform for exploring nonlinear collective behavior. Although commonly observed in mode-locked lasers, their driven dynamics remain largely unexplored, particularly the nonlinear responses of their internal motion to external modulation. Here, we present a systematic mapping of breathing soliton molecules under external modulation in the mode-locked fiber laser numerically. By tuning the driving parameters, we uncover a hierarchy of nonlinear responses that organize the internal dynamics of the molecule. The system first exhibits synchronization between the intrinsic breathing motion and the external modulation. As the driving evolves, this locking gives way to fractional entrainment at rational winding numbers. Increasing the driving strength further induces successive bifurcations, culminating in chaotic dynamics with pronounced hysteresis. Using sub-harmonic driving, we further confirm the universality of these phenomena. These results reveal the hierarchical organization of dissipative soliton molecules and therefore indicate their potential as a platform for studying universal phenomena in strongly coupled nonlinear oscillators. The ability to drive soliton molecules into targeted dynamical states may offers a route toward their application-oriented control.