Invariant Synchrony and Anti-Synchrony Subspaces of Weighted Networks

TL;DR

This paper investigates the properties and invariance of synchrony and anti-synchrony subspaces in weighted coupled cell networks, providing counts and conditions for their invariance, and confirming a conjecture in difference-coupled systems.

Contribution

It characterizes invariant polydiagonal subspaces in weighted networks, especially evenly tagged anti-synchrony subspaces, and proves a conjecture for difference-coupled graph systems.

Findings

Invariant subspaces are either synchrony or evenly tagged anti-synchrony.

Counted the number of such subspaces in weighted networks.

Confirmed a conjecture about difference-coupled graph systems.

Abstract

The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks. In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems.