  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.
