Reduced Lattices of Synchrony Subspaces and their Indices

TL;DR

This paper introduces a method to reduce the lattice of synchrony subspaces in regular coupled cell networks, enabling easier bifurcation analysis through an integer index derived from eigenvalue considerations.

Contribution

It presents a novel reduction technique for the lattice of synchrony subspaces using integer tuples, simplifying bifurcation analysis in regular networks.

Findings

Reduced lattice of synchrony subspaces obtained

Integer index facilitates bifurcation analysis

Applicable to networks with simple eigenvalues

Abstract

For a regular coupled cell network, synchrony subspaces are the polydiagonal subspaces that are invariant under the network adjacency matrix. The complete lattice of synchrony subspaces of an $n$-cell regular network can be seen as an intersection of the partition lattice of $n$ elements and a lattice of invariant subspaces of the associated adjacency matrix. We assign integer tuples with synchrony subspaces, and use them for identifying equivalent synchrony subspaces to be merged. Based on this equivalence, the initial lattice of synchrony subspaces can be reduced to a lattice of synchrony subspaces which corresponds to a simple eigenvalue case discussed in our previous work. The result is a reduced lattice of synchrony subspaces, which affords a well-defined non-negative integer index that leads to bifurcation analysis in regular coupled cell networks.