Quiver representations and dimension reduction in dynamical systems

Eddie Nijholt; Bob W. Rink; Sören Schwenker

doi:10.1137/20M1345670

Quiver representations and dimension reduction in dynamical systems

Eddie Nijholt, Bob W. Rink, Sören Schwenker

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

136 Downloads (Pure)

Abstract

Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. 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.

Original language	English
Pages (from-to)	2428-2468
Number of pages	41
Journal	SIAM Journal on Applied Dynamical Systems
Volume	19
Issue number	4
Early online date	2 Nov 2020
DOIs	https://doi.org/10.1137/20M1345670
Publication status	Published - Dec 2020

Funding

\ast Received by the editors June 15, 2020; accepted for publication (in revised form) by M. Golubitsky September 17, 2020; published electronically November 2, 2020. https://doi.org/10.1137/20M1345670 Funding: This work was partially supported by the Dutch Research Council (NWO) via the first author's research program ``Designing Network Dynamical Systems through Algebra."" The work of the second author was supported by the Sydney Mathematical Research Institute. \dagger Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (eddie.nijholt@ gmail.com). \ddagger Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, 1081 HV, The Netherlands ([email protected]). \S Department of Mathematics, Universi\a"t Hamburg, Hamburg, 22111, Germany ([email protected]).

Keywords

Bifurcation theory
Coupled networks
Normal forms
Quiver representations
Symmetry

Access to Document

10.1137/20M1345670

Quiver representations and dimension reduction in dynamical systemsFinal published version, 563 KBLicence: Article 25fa Dutch Copyright Act

Persistent URL (handle)

Persistent link

Cite this

@article{c9c757d5f8a14f4494bab4db2a90bcd6,

title = "Quiver representations and dimension reduction in dynamical systems",

abstract = "Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. 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.",

keywords = "Bifurcation theory, Coupled networks, Normal forms, Quiver representations, Symmetry",

author = "Eddie Nijholt and Rink, {Bob W.} and S{\"o}ren Schwenker",

year = "2020",

month = dec,

doi = "10.1137/20M1345670",

language = "English",

volume = "19",

pages = "2428--2468",

journal = "SIAM Journal on Applied Dynamical Systems",

issn = "1536-0040",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "4",

}

TY - JOUR

T1 - Quiver representations and dimension reduction in dynamical systems

AU - Nijholt, Eddie

AU - Rink, Bob W.

AU - Schwenker, Sören

PY - 2020/12

Y1 - 2020/12

N2 - Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. 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.

AB - Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. 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.

KW - Bifurcation theory

KW - Coupled networks

KW - Normal forms

KW - Quiver representations

KW - Symmetry

UR - http://www.scopus.com/inward/record.url?scp=85096750982&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85096750982&partnerID=8YFLogxK

U2 - 10.1137/20M1345670

DO - 10.1137/20M1345670

M3 - Article

AN - SCOPUS:85096750982

SN - 1536-0040

VL - 19

SP - 2428

EP - 2468

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 4

ER -

Quiver representations and dimension reduction in dynamical systems

Abstract

Funding

Keywords

Access to Document

Persistent URL (handle)

Other files and links

Fingerprint

Cite this