Robustness of mixed population under heterogeneity

Aging transition is a nonlinear phenomenon, which refers to the loss of macroscopic dynamical activity of a network by the stabilization of the homogeneous steady state of the network due to some kind of deterioration of the local nodes. The critical fraction of the local inactive nodes that facilitates the onset of the aging transition is recognized as a measure of the robustness of the network. Numerous efforts have been made to increase the robustness of the macroscopic oscillatory state. In this regard, we introduce a low-pass filter in the mean-field extrinsic variable of mixed population of globally coupled Stuart–Landau oscillators along with a limiting factor in the diffusive coupling to unravel the effect of their trade-off on the macroscopic oscillatory state. We also deduce the governing equation of motion for the macroscopic order parameters using a self-consistent mean-field approach. We find that decreasing the mean of the distribution of the Hopf bifurcation parameter below the null value necessitates a large standard deviation for the aging transition to occur due to an increase in the fraction of non-self-oscillatory units. Further, a large standard deviation requires a large coupling strength to facilitate the onset of the aging transition, elucidating the fact that large heterogeneity results in a more robust mixed population. Increase in the cut-off frequency of the low-pass filter and even a feeble decrease in the limiting factor favors the macroscopic oscillatory state to a large region of the parameter space despite the presence of a large fraction of non-self-oscillatory nodes. The mean-field intensity parameter facilitates the on- set of the aging transition, even at a low value of the critical fraction of non-self-oscillatory elements leading to a more fragile network. The analytical Hopf bifur- cation curve, deduced from the evolution equations for the macroscopic order parameters in the phase diagram matches with the simulation results.