On network dynamical systems with a nilpotent singularity

TL;DR

This paper explores nilpotent equilibria in network dynamical systems, demonstrating that the blow-up technique effectively analyzes these degenerate points while preserving network structure, with applications to various systems.

Contribution

It extends the blow-up technique to network systems, showing its effectiveness in desingularizing nilpotent equilibria while maintaining network structure.

Findings

Blow-up technique is suitable for network dynamical systems.

The technique preserves network structure during desingularization.

Applications include diffusive systems, internal dynamics, and Kuramoto oscillators.

Abstract

Network dynamics is nowadays of extreme relevance to model and analyze complex systems. From a dynamical systems perspective, understanding the local behavior near equilibria is of utmost importance. In particular, equilibria with at least one zero eigenvalue play a crucial role in bifurcation analysis. In this paper, we want to shed some light on nilpotent equilibria of network dynamical systems. As a main result, we show that the blow-up technique, which has proven to be extremely useful in understanding degenerate singularities in low-dimensional ordinary differential equations, is also suitable in the framework of network dynamical systems. Most importantly, we show that the blow-up technique preserves the network structure. The further usefulness of the blow-up technique, especially with regard to the desingularization of a nilpotent point, is showcased through several examples including linear diffusive systems, systems with nilpotent internal dynamics, and an adaptive network of Kuramoto oscillators.