# Networks with asymmetric inputs: lattice of synchrony subspaces

## TL;DR

This paper analyzes the structure of synchrony subspaces in coupled cell networks with asymmetric inputs, providing a detailed characterization for 1-input regular networks and methods to construct the lattice for complex networks.

## Contribution

It introduces a detailed description of join-irreducible synchrony subspaces for 1-input regular networks and offers a procedure to determine the lattice of synchrony subspaces for homogeneous networks with asymmetric inputs.

## Key findings

- Characterization of join-irreducible synchrony subspaces via eigenvectors
- Method to construct the lattice of synchrony subspaces for network unions
- Lattice of a homogeneous network as intersection of subnetworks' lattices

## Abstract

We consider coupled cell networks with asymmetric inputs and study their lattice of synchrony subspaces. For the particular case of 1-input regular coupled cell networks we describe the join-irreducible synchrony subspaces for their lattice of synchrony subspaces, first in terms of the eigenvectors and generalized eigenvectors that generate them, and then by giving a characterization of the possible patterns of the associated balanced colourings. The set of the join-irreducible synchrony subspaces is join-dense for the lattice, that is, the lattice can be obtained by sums of those join-irreducible elements (M. Aguiar, P. Ashwin, A. Dias, and M. Field. Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation, {\em J. Nonlinear Sci.} {\bf 21} (2) (2011) 271--323), and we conclude about the possible patterns of balanced colourings associated to the synchrony subspaces in the lattice. We also consider the disjoint union of two regular coupled cell networks with the same cell-type and the same edge-type. We show how to obtain the lattice of synchrony subspaces for the network union from the lattice of synchrony subspaces for the component networks. The lattice of synchrony subspaces for a homogeneous coupled cell network is given by the intersection of the lattice of synchrony subspaces for its identical-edge subnetworks per each edge-type (M. A. D. Aguiar and A. P. S. Dias. The lattice of synchrony subspaces of a coupled cell network: Characterization and computation algorithm, {\em Journal of Nonlinear Science}, {\bf 24} (6) (2014), 949--996). This, together with the results in this paper, on the lattice of synchrony subspaces for 1-input regular networks and on the lattice of synchrony subspaces for the disjoint union of networks, define a procedure to obtain the lattice of synchrony subspaces for homogeneous coupled cell networks with asymmetric inputs.