Regular synchrony lattices for product coupled cell networks.

Area of Science:

• Dynamical systems theory
• Network science
• Graph theory

Background:

• Networks are often constructed from smaller subnetworks.
• Cartesian and Kronecker products are key methods for network construction.
• Understanding synchrony subspaces is crucial for analyzing coupled dynamical systems.

Purpose of the Study:

• To investigate the relationship between synchrony subspace lattices of product networks and their component networks.
• To determine how to derive synchrony subspaces for Cartesian and Kronecker product networks.
• To compare the synchrony structures of Cartesian and Kronecker product networks.

Main Methods:

• Analysis of synchrony subspace lattices for Cartesian and Kronecker product networks.
• Mathematical proofs to establish relationships between subspace structures.
• Comparison of network construction methods and their impact on dynamics.

Main Results:

• A tensor of subspaces is a synchrony subspace for a product network if and only if the individual subspaces are synchrony subspaces for the component networks.
• Irregular synchrony subspaces for product networks may not be derivable from component networks.
• An inclusion relation exists between the synchrony subspace lattices of Cartesian and Kronecker product networks.

Conclusions:

• The lattice of regular synchrony subspaces for product networks can be constructed from component networks.
• Not all synchrony subspaces of product networks can be obtained from component networks.
• Cartesian and Kronecker products yield distinct network structures with comparable synchrony lattice relationships.