Benjamin-Feir instabilities on directed networks

TL;DR

This paper investigates Benjamin-Feir type instabilities in directed networks of coupled oscillators, revealing how network asymmetry influences the spectrum of the Laplacian and leads to new topological instabilities.

Contribution

It introduces a generalized instability framework for directed networks, extending classical Benjamin-Feir instability analysis to complex, asymmetric topologies.

Findings

Analytical characterization of instability domains for circulant networks

Identification of topological instability in directed, balanced graphs

Development of a self-consistent theory for pattern emergence

Abstract

The Complex Ginzburg-Landau equation is studied assuming a directed network of coupled oscillators. The asymmetry makes the spectrum of the Laplacian operator complex, and it is ultimately responsible for the onset of a generalized class of topological instability, reminiscent of the Benjamin-Feir type. The analysis is initially carried out for a specific class of networks, characterized by a circulant adjacency matrix. This allows us to delineate analytically the domain in the parameter space for which the generalized instability occurs. We then move forward to considering the family of non linear oscillators coupled via a generic direct, though balanced, graph. The characteristics of the emerging patterns are discussed within a self-consistent theoretical framework.