Dynamics on Networks of Manifolds

TL;DR

This paper defines continuous-time dynamical systems composed of interconnected subsystems on manifolds, using graph fibrations to relate graph structures to system properties, enabling analysis of invariant subsystems and projections.

Contribution

It introduces a rigorous framework linking graph fibrations to dynamical system maps, clarifying how graph structure influences system invariants and projections.

Findings

Graph fibrations induce maps between dynamical systems.

Surjective fibrations lead to invariant subsystems.

Injective fibrations correspond to system projections.

Abstract

We propose a precise definition of a continuous time dynamical system made up of interacting open subsystems. The interconnections of subsystems are coded by directed graphs. We prove that the appropriate maps of graphs called graph fibrations give rise to maps of dynamical systems. Consequently surjective graph fibrations give rise to invariant subsystems and injective graph fibrations give rise to projections of dynamical systems.