Coupled cell networks: semigroups, Lie algebras and normal forms

TL;DR

This paper introduces semigroup coupled cell networks, showing their vector fields form a Lie algebra, and demonstrates how to compute their normal forms, preserving symmetries and solutions near equilibria.

Contribution

It establishes the Lie algebra structure of semigroup network vector fields and extends normal form theory to these networks, including nonhomogeneous cases.

Findings

Semigroup network vector fields form a Lie algebra.

Normal forms of semigroup networks are themselves semigroup networks.

Extension to nonhomogeneous networks with semigroupoid structure.

Abstract

We introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra. This implies that near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself. Networks without the semigroup property will support normal forms with a more general network architecture, but these normal forms nevertheless possess the same symmetries and synchronous solutions as the original network. We explain how to compute Lie brackets and normal forms of coupled cell networks and we characterize the SN-decomposition that determines the normal form symmetry. This paper concludes with a generalization to nonhomogeneous networks with the structure of a semigroupoid.