Special Jordan subspaces in Coupled Cell Networks

TL;DR

This paper introduces special Jordan subspaces in regular coupled cell networks and demonstrates their fundamental role in characterizing synchrony subspaces and the lattice structure of all such subspaces.

Contribution

It defines special Jordan subspaces for regular networks and establishes their key role in describing synchrony phenomena and the lattice structure of synchrony subspaces.

Findings

Synchrony subspaces are polydiagonals that are direct sums of special Jordan subspaces.

Every join-irreducible element in the lattice contains a special Jordan subspace.

Special Jordan subspaces are central to understanding the lattice of synchrony subspaces.

Abstract

Given a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type), we define the special Jordan subspaces to that network and we use these subspaces to study the synchrony phenomenon in the theory of coupled cell networks. To be more precise, we prove that the synchrony subspaces of a regular network are precisely the polydiagonals that are direct sums of special Jordan subspaces. We also show that special Jordan subspaces play a special role in the lattice structure of all synchrony subspace because every join-irreducible element of the lattice is the smallest synchrony subspace containing some special Jordan subspace.