Robust network of globally coupled heterogeneous limit cycle oscillators due to inertia

We investigate the phase transition from macroscopic oscillatory state to stable homogeneous steady state in a heterogeneous network of globally coupled Stuart–Landau limit cycle oscillators in the presence of the inertial effect. The phase transition, known as aging transition, onsets above a critical fraction of inactive constituents in the mixed population of active and inactive units. We show that even a feeble increase in the inertial strength increases the critical fraction of inactive units significantly for the onset of the phase transition to the macroscopic steady state thereby resulting in a more robust network, in general. In contrast, a large coupling strength, in the case of a homogeneous network, facilitates the manifestation of the phase transition even for a small fraction of inactive oscillators leading to a more fragile network. Nevertheless, a large coupling strength, in the case of a heterogeneous network, increases the resilience of the network by facilitating the phase transition at a large fraction of inactive oscillators. Furthermore, a larger standard deviation of the natural frequencies always leads to a more fragile network. We derive the macroscopic evolution equations for the order parameters and the stability curve using the first-order moment expansion around the mean-field. In addition, we also deduce the critical fraction of inactive units and the critical inertial strength analytically that matches with the simulation results. Interestingly, we find that the critical inertial strength is reciprocally related to the square of the mean frequency of the network.