Haibo Ruan

Evolution of synchrony under combination of coupled cell networks

A natural way of modelling large coupled cell networks is to combine smaller networks through binary network operations. In this paper, we consider several non-product binary operations on networks such as join and coalescence, and examine the evolution of the lattice of synchrony subspaces under these operations. Classification results are obtained for synchrony subspaces of the combined network, which clarify the relation between the lattice of synchrony subspaces of the combined network and its components. Yet, in the case when the initial networks have the same edge type, this classification only applies to those synchrony subspaces that are compatible with respect to the considered operation. Based on the classification results, we give examples to show how the lattice of synchrony subspaces of the combined network can be reconstructed using the initial ones. Also, we show how the classification results can be applied to analyse the evolutionary fitness of synchrony patterns.

Interior Symmetries and Multiple Eigenvalues for Homogeneous Networks

We analyze the impact of interior symmetries on the multiplicity of the eigenvalues of the Jacobian matrix at a fully synchronous equilibrium for the coupled cell systems associated to homogeneous networks. We consider also the special cases of regular and uniform networks. It follows from our results that the interior symmetries, as well as the reverse interior symmetries and quotient interior symmetries, of the network force the existence of eigenvalues with algebraic multiplicity greater than one. The proofs are based on the special form of the adjacency matrices of the networks induced by these interior symmetries.

Synchrony and Elementary Operations on Coupled Cell Networks

Given a finite graph (network), let every node (cell) represent an individual dynamics given by a system of ordinary differential equations, and every arrow (edge) encode the dynamical influence of the tail node on the head node. We have then defined a coupled cell system that is associated with the given network structure. Subspaces that are defined by equalities of cell coordinates and left invariant under every coupled cell system respecting the network structure are called synchrony subspaces. They are completely determined by the network structure and form a complete lattice under set inclusions. We analyze the transition of the lattice of synchrony subspaces of a network that is caused by structural changes in the network topology, such as deletion and addition of cells or edges, and rewirings of edges. We give sufficient, and in some cases both sufficient and necessary, conditions under which lattice elements persist or disappear.

Symmetry Analysis of Coupled Scalar Systems under Time Delay

We study systems of coupled units in a general network configuration with a coupling delay. We show that the destabilizing bifurcations from an equilibrium are governed by the extreme eigenvalues of the coupling matrix of the network. Based on the equivariant degree method and its computational packages, we perform a symmetry classification of destabilizing bifurcations in bidirectional rings of coupled units. Both stationary and oscillatory bifurcations are discussed. We also introduce the concept of secondary dominating orbit types to capture bifurcating solutions of submaximal nature.

A degree theory for coupled cell systems with quotient symmetries

We introduce a topological degree theory for the study of Hopf bifurcations in coupled cell systems whose quotient systems (obtained by restricting the system to its flow-invariant subspaces) possess various symmetries. To describe the structure of these quotient symmetries, we introduce the concept of a representation lattice, which is defined as a lattice of representation spaces of (different) symmetry groups that satisfy a compatibility and a consistence condition. Based on the (twisted) equivariant degree, we define a lattice-equivariant degree for maps that are compatible with respect to this representation lattice structure. We apply the lattice-equivariant degree to study a synchrony-breaking Hopf-bifurcation problem in (homogeneous) coupled cell systems and obtain a topological classification of all bifurcating branches of oscillating solutions according to their synchrony types and their symmetric properties.