Networks of open systems

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 (Spivak, 2013; Rupel and Spivak, 2013; Vagner et al., 2015). It is also very useful to consider maps between dynamical systems. This is the point of view taken by DeVille and Lerman in the study of dynamics on networks (DeVille and Lerman, 2015 14,51; DeVille and Lerman, 2010). The work of DeVille and Lerman was inspired by the coupled cell networks of Golubitsky, Stewart and their collaborators (Stewart et al., 2003; Golubitsky et al., 2005; Golubitsky and Stewart, 2006). The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. More specifically we define a double category of open systems and construct a functor from this double category to the double category of vector spaces, linear maps and linear relations. This allows us, on one hand, to build new open systems out of collections of smaller open subsystems and on the other to 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. (C) 2018 Elsevier B.V. All rights reserved.