Interplay between network topology and synchrony-breaking bifurcation: homogeneous four-cell coupled networks

Abstract

Complex networks are studied across many fields of science. Much progress has been made on static and statistical features of networks, such as small world and scale-free networks. However, general studies of network dynamics are comparatively rare. Synchrony is one commonly observed dynamical behaviour in complex networks. Synchrony breaking is where a fully synchronised network loses coherence, and breaks up into multiple clusters of self-synchronised sub-networks. Mathematically this can be described as a bifurcation from a fully synchronous state, and in this thesis we investigate the effect of network topology on synchrony-breaking bifurcations. Coupled cell networks represent a collection of individual dynamical systems (termed cells) that interact with each other. Each cell is described by an ordinary differential equation (ODE) or a system of ODEs. Schematically, the architecture of a coupled cell network can be represented by a directed graph with a node for each cell, and edges indicating cell couplings. Regular homogeneous networks are a special case where all the nodes/cells and edges are of the same type, and every node has the same number of input edges, which we call the valency of the network. Classes of homogeneous regular networks can be counted using an existing group theoretic enumeration formula, and this formula is extended here to enumerate networks with more generalised structures. However, this does not generate the networks themselves. We therefore develop a computer algorithm to display all connected regular homogeneous networks with less than six cells and analysed synchrony-breaking bifurcations for four-cell regular homogeneous networks. Robust patterns of synchrony (invariant synchronised subspaces under all admissible vector fields) describe how cells are divided into multiple synchronised clusters, and their existence is solely determined by the network topology. These robust patterns of synchrony have a hierarchical relationship, and can be treated as a partially ordered set, and expressed as a lattice. For each robust pattern of synchrony (or lattice point) we can reduce the original network to a smaller network, called a quotient network, by representing each cluster as a single combined node. Therefore, the lattice for a given regular homogeneous network provides robust patterns of synchrony and corresponding quotient networks. Some lattice structures allow a synchrony breaking bifurcation analysis based solely on the dynamics of the quotient networks, which are lifted to the original network using the robust patterns of synchrony. However, in other cases the lattice structure also tells us of the existence and location of additional synchrony-breaking bifurcating branches not seen in the quotient networks. In conclusion the work undertaken here shows that the invariant synchronised subspaces that arise from a network topology facilitate the classification of synchrony-breaking bifurcations of networks