Linear dynamical systems on graphs

TL;DR

This paper studies linear dynamical systems structured on multigraphs, analyzing their stability, growth, and stabilizability, with methods for invariant multinorms and applications to fractals, attractors, and numerical methods.

Contribution

It generalizes classical linear switching systems to multigraph structures and provides a method to construct invariant multinorms for irreducible systems.

Findings

Existence of invariant multinorms for strongly connected multigraph systems

Method to compute joint spectral radius and Lyapunov exponent

Applications demonstrated in fractals, attractors, and ODE multistep methods

Abstract

We consider linear dynamical systems with a structure of a multigraph. The vertices are associated to linear spaces and the edges correspond to linear maps between those spaces. We analyse the asymptotic growth of trajectories (associated to paths along the multigraph), the stability and the stabilizability problems. This generalizes the classical linear switching systems and their recent extensions to Markovian systems, to systems generated by regular languages, etc. We show that an arbitrary system can be factorized into several irreducible systems on strongly connected multigraphs. For the latter systems, we prove the existence of invariant (Barabanov) multinorm and derive a method of its construction. The method works for a vast majority of systems and finds the joint spectral radius (Lyapunov exponent). Numerical examples are presented and applications to the study of fractals, attractors, and multistep methods for ODEs are discussed.