Dynamical systems defined on simplicial complexes: symmetries, conjugacies, and invariant subspaces

TL;DR

This paper explores dynamical systems on simplicial complexes, analyzing how symmetries influence system behavior, conjugacy classes, and invariant subspaces, providing a framework for understanding complex network dynamics.

Contribution

It characterizes conjugacy classes and demonstrates how symmetries in simplicial complexes affect the dynamics and invariant subspaces, offering new insights into their structure.

Findings

Symmetries induce specific invariant subspaces in the dynamics.

Conjugacy classes are characterized for systems on simplicial complexes.

The framework links complex symmetries to dynamical behavior.

Abstract

We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.