Nonlinear Diffusion on Networks: Perturbations and Consensus Dynamics

TL;DR

This paper investigates nonlinear diffusion equations on networks, analyzing equilibrium structures, singularities, and consensus dynamics, with a focus on perturbations, slow-fast systems, and canard solutions, supported by numerical simulations.

Contribution

It characterizes equilibrium sets and singularities in nonlinear network diffusion, introduces analysis of perturbations and canards, and extends findings to non-complete graphs.

Findings

Singularities on the consensus space are generically transcritical.

Existence of canard solutions under local assumptions.

Numerical simulations validate theoretical results and reveal transient patterns.

Abstract

In this paper, we study a class of equations representing nonlinear diffusion on networks. A particular instance of our model can be seen as a network equivalent of the porous-medium equation. We are interested in studying perturbations of such a system and describing the consensus dynamics. The nonlinearity of the equations gives rise to potentially intricate structures of equilibria that can intersect the consensus space, creating singularities. For the unperturbed case, we characterise the sets of equilibria by exploiting the symmetries under group transformations of the nonlinear vector field. Under small perturbations, we obtain a slow-fast system. Thus, we analyse the slow-fast dynamics near the singularities on the consensus space. The analysis at this stage is carried out for complete networks, allowing a detailed characterisation of the system. We provide a linear approximation of the intersecting branches of equilibria at the singular points; as a consequence, we show that, generically, the singularities on the consensus space turn out to be transcritical. We prove under local assumptions the existence of canard solutions. For generic graph structures, assuming more strict conditions on the perturbation, we prove the existence of a maximal canard, which coincides with the consensus subspace. In addition, we validate by numerical simulations the principal findings of our main theory, extending the study to non-complete graphs. Moreover, we show how the delayed loss of stability associated to the canards induces transient spatio-temporal patterns.