# Phase reduction beyond the first order: the case of the mean-field complex Ginzburg-Landau equation

## TL;DR

This paper develops a second-order phase reduction method for the mean-field complex Ginzburg-Landau equation, capturing complex collective dynamics like chaos that first-order models miss.

## Contribution

It introduces an isochron-based scheme for second-order phase approximation, extending to third order, revealing multi-body interactions in oscillator ensembles.

## Key findings

- Second-order approximation reproduces weak coupling dynamics of MF-CGLE.
- Higher-order terms introduce multi-body interactions.
- Multi-body interactions are linked to collective chaos.

## Abstract

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.