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.

中文翻译：

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由于惯性而产生的全局耦合异构极限环振荡器的鲁棒网络

我们研究了在存在惯性效应的情况下，全局耦合斯图尔特-朗道极限环振荡器的异构网络中从宏观振荡状态到稳定均质稳态的相变。相变（称为老化转变）在活性和非活性单元的混合群体中非活性成分的临界比例之上开始。我们表明，即使惯性强度的微弱增加，也会显着增加非活动单元的临界分数，从而导致相变到宏观稳态的开始，从而总体上产生更稳健的网络。相反，在同质网络的情况下，大的耦合强度即使对于一小部分不活动的振荡器也有利于相变的表现，从而导致网络更加脆弱。然而，在异构网络的情况下，大的耦合强度通过促进大部分不活动振荡器的相变来增加网络的弹性。此外，固有频率的较大标准偏差总是导致网络更加脆弱。我们使用平均场周围的一阶矩展开推导了阶次参数和稳定性曲线的宏观演化方程。此外，我们还通过分析推导了与仿真结果相匹配的非活动单元的临界比例和临界惯性强度。有趣的是，我们发现临界惯性强度与网络平均频率的平方成反比。