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

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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.

Funding

˚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]).

Keywords

Computer-assisted proofs
Continuation
Desingularization
Hopf bifurcation

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10.1137/20M1343464

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

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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",

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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",

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doi = "10.1137/20M1343464",

language = "English",

volume = "20",

pages = "573--607",

journal = "SIAM Journal on Applied Dynamical Systems",

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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

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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

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