# Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

## TL;DR

This paper derives a Ginzburg-Landau approximation for reaction-diffusion systems on directed graphs near a Hopf bifurcation, enabling analysis of stability and pattern formation in coupled oscillators.

## Contribution

It introduces a complex Ginzburg-Landau equation tailored for directed networks, linking network topology with oscillatory stability analysis.

## Key findings

- The CGLE accurately predicts stability and symmetry-breaking instabilities.
- Numerical simulations with the Brusselator model validate the theoretical predictions.
- Pattern formations from the CGLE match those from the original reaction-diffusion system.

## Abstract

A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg-Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.