High-order phase reduction for coupled 2D oscillators

Phase reduction is a general approach to describe coupled oscillatory units in terms of their phases, assuming that the amplitudes are enslaved. The coupling should be small for such reduction, but one also expects the reduction to be valid for finite coupling. This paper presents a general framework, allowing us to obtain coupling terms in higher orders of the coupling parameter for generic two-dimensional oscillators and arbitrary coupling terms. The theory is illustrated with an accurate prediction of Arnold's tongue for the van der Pol oscillator exploiting higher-order phase reduction.