On network dynamical systems with a nilpotent singularity

Hildeberto Jardón-Kojakhmetov, Christian Kuehn

Published 2023-10-13Version 1

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.