Heterogeneous nucleation due to adaptation disparity and phase lag in a finite-size adaptive dynamical network

We investigate the influence of adaptation disparity and phase lag on the heterogeneous nucleation leading to multi-step and single-step phase transitions in a finite-size adaptive network due to frequency disorders. We elucidate that the adaptation disparity predominates the effect of frequency disorder in seeding the heterogeneous nucleation. Further, a large phase lag facilitates almost a continuous spectrum of partially and fully synchronized frequency clusters, and multistability among them. We also find that the underlying mechanisms of heterogeneous nucleation due to the adaptation disparity and phase lag parameters are distinctly different from that observed only in the presence of disorders. Furthermore, we also analytically deduce the macroscopic evolution equations for the cluster dynamics using the collective coordinates framework and show that resulting reduced dynamics mimic the observed heterogeneous phase transitions of the adaptive network under adaptation disparity and phase lag. Further, we deduce the upper bound for the coupling strength for the existence of two intra-clusters explicitly in terms of the adaptation disparity and phase lag parameters for the onset of the abrupt single-step transition.