Stabilization of hyperbolic reaction-diffusion systems on directed networks through the generalized Routh-Hurwitz criterion for complex polynomials

TL;DR

This paper extends the Routh-Hurwitz criterion to analyze the stability of hyperbolic reaction-diffusion systems on directed networks with complex coefficients, using the Brusselator model as a case study.

Contribution

It introduces a generalized Routh-Hurwitz criterion for complex polynomials to assess stability in networked systems with asymmetric couplings.

Findings

Identifies stability regions for the Brusselator model on directed networks.

Shows how complex coefficient polynomials affect system stability.

Provides a new tool for stability analysis of complex networked systems.

Abstract

The study of dynamical systems on complex networks is of paramount importance in engineering, given that many natural and artificial systems find a natural embedding on discrete topologies. For instance, power grids, chemical reactors and the brain, to name a few, can be modeled through the network formalism by considering elementary units coupled via the links. In recent years, scholars have developed numerical and theoretical tools to study the stability of those coupled systems when subjected to perturbations. In such framework, it was found that asymmetric couplings enhance the possibilities for such systems to become unstable. Moreover, in this scenario the polynomials whose stability needs to be studied bear complex coefficients, which makes the analysis more difficult. In this work, we put to use a recent extension of the well-known Routh-Hurwitz stability criterion, allowing to treat the complex coefficient case. Then, using the Brusselator model as a case study, we discuss the stability conditions and the regions of parameters when the networked system remains stable.