Quiver representations and dimension reduction in dynamical systems

TL;DR

This paper demonstrates how quiver representations can encode geometric properties of dynamical systems, and shows these structures are preserved under common reduction techniques, aiding in the analysis of complex systems.

Contribution

It introduces the use of quiver representations to encode and analyze geometric properties of dynamical systems, and proves their invariance under reduction methods.

Findings

Quiver structures encode symmetry and network relations in dynamical systems.

Quiver equivariance is preserved under Lyapunov-Schmidt, center manifold, and normal form reductions.

Provides a framework for analyzing complex dynamical systems with geometric properties.

Abstract

Dynamical systems often admit geometric properties that must be taken into account when studying their behaviour. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov-Schmidt reduction, center manifold reduction, and normal form reduction.