Rigorous verification of Hopf bifurcations via desingularization and continuation

Jan Bouwe Van den Berg; Jean Philippe Lessard; Elena Queirolo

doi:10.1137/20M1343464

Rigorous verification of Hopf bifurcations via desingularization and continuation

Jan Bouwe Van den Berg, Jean Philippe Lessard, Elena Queirolo

Mathematics

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

114 Downloads (Pure)

Abstract

In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddlenode bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems.

Original language	English
Pages (from-to)	573-607
Number of pages	35
Journal	SIAM Journal on Applied Dynamical Systems
Volume	20
Issue number	2
Early online date	1 Apr 2021
DOIs	https://doi.org/10.1137/20M1343464
Publication status	Published - Apr 2021

Bibliographical note

Funding Information:
˚Received by the editors June 5, 2020; accepted for publication (in revised form) February 10, 2021; published electronically April 1, 2021. https://doi.org/10.1137/20M1343464 Funding: The work of the first author was partially supported by NWO-VICI grant 639033109. The work of the second author was supported by NSERC. :Department of Mathematics, VU Amsterdam, 1081 HV Amsterdam, The Netherlands ([email protected]). ;Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St W, Montreal, QC, H3A 0B9, Canada ([email protected]). §Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019 USA ([email protected]).

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.

Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

Keywords

Computer-assisted proofs
Continuation
Desingularization
Hopf bifurcation

Access to Document

10.1137/20M1343464

Rigorous verification of Hopf bifurcations via desingularization and continuationFinal published version, 661 KBLicence: Article 25fa Dutch Copyright Act

Persistent URL (handle)

Persistent link

Cite this

@article{916975b198fa497e8e2cb3deee2fb274,

title = "Rigorous verification of Hopf bifurcations via desingularization and continuation",

abstract = "In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddlenode bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems. ",

keywords = "Computer-assisted proofs, Continuation, Desingularization, Hopf bifurcation",

author = "{Van den Berg}, {Jan Bouwe} and Lessard, {Jean Philippe} and Elena Queirolo",

note = "Funding Information: ˚Received by the editors June 5, 2020; accepted for publication (in revised form) February 10, 2021; published electronically April 1, 2021. https://doi.org/10.1137/20M1343464 Funding: The work of the first author was partially supported by NWO-VICI grant 639033109. The work of the second author was supported by NSERC. :Department of Mathematics, VU Amsterdam, 1081 HV Amsterdam, The Netherlands ([email protected]). ;Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St W, Montreal, QC, H3A 0B9, Canada ([email protected]). §Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019 USA ([email protected]). Publisher Copyright: {\textcopyright} 2021 Society for Industrial and Applied Mathematics. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",

year = "2021",

month = apr,

doi = "10.1137/20M1343464",

language = "English",

volume = "20",

pages = "573--607",

journal = "SIAM Journal on Applied Dynamical Systems",

issn = "1536-0040",

publisher = "Society of Industrial and Applied Mathematics",

number = "2",

}

TY - JOUR

T1 - Rigorous verification of Hopf bifurcations via desingularization and continuation

AU - Van den Berg, Jan Bouwe

AU - Lessard, Jean Philippe

AU - Queirolo, Elena

N1 - Funding Information: ˚Received by the editors June 5, 2020; accepted for publication (in revised form) February 10, 2021; published electronically April 1, 2021. https://doi.org/10.1137/20M1343464 Funding: The work of the first author was partially supported by NWO-VICI grant 639033109. The work of the second author was supported by NSERC. :Department of Mathematics, VU Amsterdam, 1081 HV Amsterdam, The Netherlands ([email protected]). ;Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St W, Montreal, QC, H3A 0B9, Canada ([email protected]). §Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019 USA ([email protected]). Publisher Copyright: © 2021 Society for Industrial and Applied Mathematics. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/4

Y1 - 2021/4

N2 - In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddlenode bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems.

AB - In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddlenode bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems.

KW - Computer-assisted proofs

KW - Continuation

KW - Desingularization

KW - Hopf bifurcation

UR - http://www.scopus.com/inward/record.url?scp=85105313295&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85105313295&partnerID=8YFLogxK

U2 - 10.1137/20M1343464

DO - 10.1137/20M1343464

M3 - Article

AN - SCOPUS:85105313295

SN - 1536-0040

VL - 20

SP - 573

EP - 607

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 2

ER -

Rigorous verification of Hopf bifurcations via desingularization and continuation

Abstract

Bibliographical note

Keywords

Access to Document

Persistent URL (handle)

Other files and links

Fingerprint

Cite this