  Phase reduction is a powerful technique that makes possible describe the
dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase.
For ensembles of oscillators, a classical example of phase reduction is the
derivation of the Kuramoto model from the mean-field complex Ginzburg-Landau
equation (MF-CGLE). Still, the Kuramoto model is a first-order phase
approximation that displays either full synchronization or incoherence, but
none of the nontrivial dynamics of the MF-CGLE. This fact calls for an
expansion beyond the first order in the coupling constant. We develop an
isochron-based scheme to obtain the second-order phase approximation, which
reproduces the weak coupling dynamics of the MF-CGLE. The practicality of our
method is evidenced by extending the calculation up to third order. Each new
term of the power series expansion contributes with additional higher-order
multi-body (i.e.non-pairwise) interactions. This points to intricate multi-body
phase interactions as the source of pure collective chaos in the MF-CGLE at
moderate coupling.
