High-order phase reduction for coupled oscillators

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 allow for the obtaining of the higher-order approximation explicitly. 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.