We explore the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter. For coupled Stuart–Landau oscillators, where the phase can be introduced explicitly, we develop an analytic perturbation procedure to explicitly obtain the higher-order approximation. We demonstrate this by deriving the second-order phase equations for a network of three Stuart–Landau oscillators. For systems where explicit expressions of the phase are not available, we present a numerical procedure that constructs the phase dynamics equations for a small network of coupled units. We apply this approach to a network of three van der Pol oscillators and reveal components in the coupling with different scaling in the interaction strength.

我们以耦合参数的较高阶次探索耦合振荡器网络中的相位降低。对于可以明确引入相位的斯图尔特-兰道耦合振荡器，我们开发了一种解析扰动程序来明确获得高阶近似。我们通过推导三个Stuart–Landau振荡器网络的二阶相位方程来证明这一点。对于没有明确的相位表达式的系统，我们提出了一种数值程序，该程序为耦合单元的小型网络构造了相位动力学方程。我们将此方法应用于由三个范德波尔振荡器组成的网络，并揭示了在相互作用强度上具有不同缩放比例的耦合分量。