Nonlinear diffusion on networks: Perturbations and consensus dynamics

In this paper, we study a class of equations representing nonlinear diffusion on networks. A particular instance of our model could 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 could intersect the consensus space, creating singularities. For the unperturbed case, we characterize 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 analyze 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 characterization 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 with the canards induces transient spatio-temporal patterns.