Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE

Jan Bouwe van den Berg; Elena Queirolo

doi:10.1016/j.cnsns.2021.106133

Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE

Jan Bouwe van den Berg, Elena Queirolo^*

^*Corresponding author for this work

Mathematics

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Abstract

We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto–Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton–Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.

Original language	English
Article number	106133
Pages (from-to)	1-22
Number of pages	22
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	108
Early online date	2 Dec 2021
DOIs	https://doi.org/10.1016/j.cnsns.2021.106133
Publication status	Published - May 2022

Bibliographical note

Funding Information:
Both authors were partially supported by NWO-VICI grant 639.033.109 , while E.Q. is also partially supported by NSF grant DMS-1839294 .

Publisher Copyright:
© 2021 Elsevier B.V.

Funding

Both authors were partially supported by NWO-VICI grant 639.033.109 , while E.Q. is also partially supported by NSF grant DMS-1839294 .

Funders	Funder number
NWO-VICI	639.033.109
National Science Foundation	DMS-1839294
National Science Foundation

Keywords

Hpof bifurcations
Nonlinear dynamics
PDEs
Validated numerics

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10.1016/j.cnsns.2021.106133

Rigorous validation of a Hopf bifurcation in the KuramotoSivashinsky PDEFinal published version, 557 KBLicence: Article 25fa Dutch Copyright Act

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title = "Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE",

abstract = "We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto–Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton–Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.",

keywords = "Hpof bifurcations, Nonlinear dynamics, PDEs, Validated numerics",

author = "{van den Berg}, {Jan Bouwe} and Elena Queirolo",

note = "Funding Information: Both authors were partially supported by NWO-VICI grant 639.033.109 , while E.Q. is also partially supported by NSF grant DMS-1839294 . Publisher Copyright: {\textcopyright} 2021 Elsevier B.V.",

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AU - van den Berg, Jan Bouwe

AU - Queirolo, Elena

N1 - Funding Information: Both authors were partially supported by NWO-VICI grant 639.033.109 , while E.Q. is also partially supported by NSF grant DMS-1839294 . Publisher Copyright: © 2021 Elsevier B.V.

PY - 2022/5

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N2 - We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto–Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton–Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.

AB - We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto–Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton–Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.

KW - Hpof bifurcations

KW - Nonlinear dynamics

KW - PDEs

KW - Validated numerics

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JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

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Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE

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