Chasing patterns: diffusion-driven and long-range instabilities in networked systems

by Riccardo Muolo (UNamur)


Many natural and artificial systems exhibit collective behavior, which show in the form of spatio-temporal patterns. This have triggered the interests ofscholar, who have proposed several theories to account for such diversity. One of the most popular mechanisms of pattern formation is due to Alan Turing, who showed that diffusion can disrupt a homogeneous stable state, triggering an instability. The original theory has been conceived in the framework of reaction-diffusion PDEs, but it has been recently extended on networked systems. Moreover, it has been shown that a Turing-like mechanism occur in the framework of synchronized coupled oscillators, as diffusion can lead to a loss of synchronization. In this seminar I will present an overview of Turing theory in networked systems, showing that the Turing framework is indeed not too far from that of coupled chaotic oscillators. I will then focus the attention on two results we have recently obtained. The first one, about the study of reaction-diffusion systems on top of non-normal networks, i.e., networks whose adjacency matrix is non-normal. Such topology makes the system more sensible to perturbation, leading to a loss of stability even when a linear stability analysis predicts otherwise. Finally, I will describe a way to obtain patterns without diffusion, as the instability is triggered by the sole non-local interactions.