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Pitchfork bifurcation and heteroclinic connections in the Kuramoto–Sivashinsky PDE
Journal
Journal of Computational Dynamics
100
Author
Kubica Jacek
Zgliczyński Piotr
Kalita Piotr
Volume
12
Number
3
Pages
325-392
eISSN
2158-2505
Keywords in English
Pitchfork bifurcation
heteroclinic connection
dissipative PDEs
Galerkin projection
computer–assisted proof
rigorous numerics
Access date
2025-08-04
Language
English
Journal language
English
Abstract in English
We present a method for the complete analysis of the dynamics of dissipative partial differential equations (PDEs) undergoing a pitchfork bifurcation. We apply our technique to the Kuramoto–Sivashinsky PDE on the line to obtain a computer-assisted proof of the creation of two symmetric branches of non-symmetric fixed points and heteroclinic connections between the symmetric fixed point and the new ones. The range of parameters is given explicitly and is large enough to allow for the rigorous continuation of the fixed points and heteroclinic connections created during the bifurcation.
Affiliation
Wydział Matematyki i Informatyki : Instytut Informatyki i Matematyki Komputerowej
| dc.abstract.en | We present a method for the complete analysis of the dynamics of dissipative partial differential equations (PDEs) undergoing a pitchfork bifurcation. We apply our technique to the Kuramoto–Sivashinsky PDE on the line to obtain a computer-assisted proof of the creation of two symmetric branches of non-symmetric fixed points and heteroclinic connections between the symmetric fixed point and the new ones. The range of parameters is given explicitly and is large enough to allow for the rigorous continuation of the fixed points and heteroclinic connections created during the bifurcation. | |
| dc.affiliation | Wydział Matematyki i Informatyki : Instytut Informatyki i Matematyki Komputerowej | |
| dc.contributor.author | Kubica, Jacek - 235432 | |
| dc.contributor.author | Zgliczyński, Piotr - 132902 | |
| dc.contributor.author | Kalita, Piotr - 128604 | |
| dc.date.accession | 2025-08-04 | |
| dc.date.accessioned | 2025-08-04T06:09:15Z | |
| dc.date.available | 2025-08-04T06:09:15Z | |
| dc.date.createdat | 2025-07-31T06:09:50Z | en |
| dc.date.issued | 2025 | |
| dc.date.openaccess | 0 | |
| dc.description.accesstime | w momencie opublikowania | |
| dc.description.number | 3 | |
| dc.description.physical | 325-392 | |
| dc.description.version | ostateczna wersja wydawcy | |
| dc.description.volume | 12 | |
| dc.identifier.doi | 10.3934/jcd.2025001 | |
| dc.identifier.eissn | 2158-2505 | |
| dc.identifier.project | UMO-2016/22/A/ST1/00077 | |
| dc.identifier.project | DRC AI | |
| dc.identifier.uri | https://ruj.uj.edu.pl/handle/item/558668 | |
| dc.language | eng | |
| dc.language.container | eng | |
| dc.rights | Udzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa | |
| dc.rights.licence | CC-BY | |
| dc.rights.uri | http://creativecommons.org/licenses/by/4.0/legalcode.pl | |
| dc.share.type | otwarte czasopismo | |
| dc.subject.en | Pitchfork bifurcation | |
| dc.subject.en | heteroclinic connection | |
| dc.subject.en | dissipative PDEs | |
| dc.subject.en | Galerkin projection | |
| dc.subject.en | computer–assisted proof | |
| dc.subject.en | rigorous numerics | |
| dc.subtype | Article | |
| dc.title | Pitchfork bifurcation and heteroclinic connections in the Kuramoto–Sivashinsky PDE | |
| dc.title.journal | Journal of Computational Dynamics | |
| dc.type | JournalArticle | |
| dspace.entity.type | Publication | en |
dc.abstract.en
We present a method for the complete analysis of the dynamics of dissipative partial differential equations (PDEs) undergoing a pitchfork bifurcation. We apply our technique to the Kuramoto–Sivashinsky PDE on the line to obtain a computer-assisted proof of the creation of two symmetric branches of non-symmetric fixed points and heteroclinic connections between the symmetric fixed point and the new ones. The range of parameters is given explicitly and is large enough to allow for the rigorous continuation of the fixed points and heteroclinic connections created during the bifurcation. dc.affiliation
Wydział Matematyki i Informatyki : Instytut Informatyki i Matematyki Komputerowej dc.contributor.author
Kubica, Jacek - 235432 dc.contributor.author
Zgliczyński, Piotr - 132902 dc.contributor.author
Kalita, Piotr - 128604 dc.date.accession
2025-08-04 dc.date.accessioned
2025-08-04T06:09:15Z dc.date.available
2025-08-04T06:09:15Z dc.date.createdaten
2025-07-31T06:09:50Z dc.date.issued
2025 dc.date.openaccess
0 dc.description.accesstime
w momencie opublikowania dc.description.number
3 dc.description.physical
325-392 dc.description.version
ostateczna wersja wydawcy dc.description.volume
12 dc.identifier.doi
10.3934/jcd.2025001 dc.identifier.eissn
2158-2505 dc.identifier.project
UMO-2016/22/A/ST1/00077 dc.identifier.project
DRC AI dc.identifier.uri
https://ruj.uj.edu.pl/handle/item/558668 dc.language
eng dc.language.container
eng dc.rights
Udzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa dc.rights.licence
CC-BY dc.rights.uri
http://creativecommons.org/licenses/by/4.0/legalcode.pl dc.share.type
otwarte czasopismo dc.subject.en
Pitchfork bifurcation dc.subject.en
heteroclinic connection dc.subject.en
dissipative PDEs dc.subject.en
Galerkin projection dc.subject.en
computer–assisted proof dc.subject.en
rigorous numerics dc.subtype
Article dc.title
Pitchfork bifurcation and heteroclinic connections in the Kuramoto–Sivashinsky PDE dc.title.journal
Journal of Computational Dynamics dc.type
JournalArticle dspace.entity.typeen
Publication Affiliations
Wydział Matematyki i Informatyki
Kubica, Jacek
mathematics
Zgliczyński, Piotr
computer and information sciences
Kalita, Piotr
mathematics
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