Chaotic behaviour in diffusively coupled systems - Queiroz, Fernando Cordeiro de (2022)
We study emergent oscillatory behaviour in networks of diffusively coupled nonlinear ordinary differential equations. Starting from a situation where the isolated dynamics at each node are the same and possess a globally attractive equilibrium point. Recent research has shown that general networks can present periodic oscillations due to diffusive coupling under mild conditions in the isolated vector field. In this thesis, we provide conditions on the isolated vector field and the underlying graph such that the network has a center manifold and we show that the reduced vector field has nonvanishing Taylor coefficients whenever the original vector field is generic. Moreover, we show that when the dimension of the isolated vector field is at least four its is possible to find positive-definite matrices serving as couplings such that the network has a nilpotent singularity which corresponds to the existence of a three-dimensional center manifold. As a consequence, the network will present a chaotic behaviour.