History dependence in synchronization from rate asymmetry and phase weight feedback in adaptive networks

Adaptive networks can exhibit pronounced dependence on dynamical history, but the mechanisms controlling the extent and organization of this dependence are not fully understood. Here we study an adaptive Kuramoto network in which the relative rates of coupling strengthening and weakening are varied while the total adaptation-rate budget is held fixed. Weakening-dominated adaptation produces a substantially broader history-dependent synchronization interval than symmetric adaptation, whereas strengthening-dominated adaptation strongly suppresses it. Cross-initialization of phase and weight histories shows that both components influence the final collective state, while a strong non-additive phase–weight interaction appears in the weakening-dominated regime. This indicates that the observed history dependence is associated with the joint organization of phases and adaptive weights rather than with either subsystem alone. Long-time simulations further show that the resulting low- and high-history regimes remain macroscopically separated over timescales much longer than the microscopic adaptation timescales. A reduced mean-field theory links coherence to the fraction of strengthening edges, yields an adaptive branch relation and fold condition, and identifies an asymmetry-dependent feedback coefficient Γ(ρ). The resulting Γ^2 scaling reproduces the observed trend in hysteresis suppression without fitting the numerical widths. Together, these results show how rate asymmetry can regulate both the extent and organization of history-dependent synchronization in adaptive networks.