Center manifolds of coupled cell networks

Dynamical systems with a network structure can display anomalous bifurcations as a generic phenomenon. As an explanation for this it has been noted that homogeneous networks can be realized as quotient networks of so-called fundamental networks. The class of admissible vector fields for these fundamental networks is equal to the class of equivariant vector fields of the regular representation of a monoid. Using this insight, we set up a framework for center manifold reduction in fundamental networks and their quotients. We then use this machinery to explain the difference in generic bifurcations between three example networks with identical spectral properties and identical robust synchrony spaces

Symmetry groupoids and patterns of synchrony in coupled cell networks

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only mechanism that can create such states in a coupled cell system and show that it is not. The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with the corresponding internal dynamics and couplings—are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is “robust” if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of “polydiagonal” subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an equivalence relation on cells is “balanced.” The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the “quotient network.” The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems

Weak chimeras in minimal networks of coupled phase oscillators

We suggest a definition for a type of chimera state that appears in networks of indistinguishable phase oscillators. Defining a "weak chimera" as a type of invariant set showing partial frequency synchronization, we show that this means they cannot appear in phase oscillator networks that are either globally coupled or too small. We exhibit various networks of four, six and ten indistinguishable oscillators where weak chimeras exist with various dynamics and stabilities. We examine the role of Kuramoto-Sakaguchi coupling in giving degenerate (neutrally stable) families of weak chimera states in these example networks.Comment: 9 figure

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

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

On Tree-Partition-Width

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph with tree-width $k\geq3$ and maximum degree $\Delta\geq1$ has tree-partition-width at most $24k\Delta$. We prove that this bound is within a constant factor of optimal. In particular, for all $k\geq3$ and for all sufficiently large $\Delta$, we construct a graph with tree-width $k$, maximum degree $\Delta$, and tree-partition-width at least (\eighth-\epsilon)k\Delta. Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}\Delta-1)$ without the restriction that $k\geq3$