# Networks of open systems

## TL;DR

This paper develops an algebraic framework for networks of open systems, enabling construction of complex systems from smaller parts and tracking maps between them, with applications to synchronization phenomena.

## Contribution

It unifies algebraic and categorical approaches to systems of systems, allowing for composition and morphisms of open systems within a single structure.

## Key findings

- Provides a new algebraic structure for open systems
- Enables construction of large systems from smaller subsystems
- Generalizes existing synchronization results

## Abstract

Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems, which in effect amounts to viewing dynamical systems as objects in an appropriate category. This is the point of view taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. Consequently we obtain synchrony results for open systems which generalize the synchrony results of Golubitsky, Stewart and their collaborators for groupoid invariant vector fields on coupled cell networks.