Formal Setting for Period Doubling Bifurcation of Limit Cycles

D 2023

Formal Setting for Period Doubling Bifurcation of Limit Cycles

ZÁTHURECKÝ, Jakub

Základní údaje

Originální název

Formal Setting for Period Doubling Bifurcation of Limit Cycles

Autoři

ZÁTHURECKÝ, Jakub

Vydání

Cham (Switzerland), 15th Chaotic Modeling and Simulation International Conference, od s. 381-395, 15 s. 2023

Nakladatel

Springer

Další údaje

Jazyk

angličtina

Typ výsledku

Stať ve sborníku

Obor

10100 1.1 Mathematics

Stát vydavatele

Švýcarsko

Utajení

není předmětem státního či obchodního tajemství

Forma vydání

tištěná verze "print"

Odkazy

URL

Označené pro přenos do RIV

Ano

Kód RIV

RIV/00216224:14310/23:00131250

Organizační jednotka

Přírodovědecká fakulta

ISBN

978-3-031-27081-9

ISSN

Klíčová slova anglicky

Limit cycle; Period doubling; Fredholm operator; Lyapunov-Schmidt reduction; Pitchfork bifurcation

Štítky

14311010, rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 22. 7. 2025 09:28, Mgr. Marie Novosadová Šípková, DiS.

Anotace

V originále

A rigorous description of period doubling bifurcation of limit cycles in autonomous systems of first order differential equations based on tools of functional analysis and singularity theory is presented. It is an alternative approach which is independent of the theory of discrete-time dynamical systems, especially Poincaré sections. Particularly, sufficient conditions for its occurrence and its normal form coefficients are expressed in terms of derivatives of the operator defining given equations. Also, stability of solutions is analysed and it is related to particular derivatives of the operator. Our approach is an adjustment of techniques used by Golubitsky and Schaeffer (Singularities and Groups in Bifurcation Theory: Volume 1. Springer, New York, 1985) in the study of Hopf bifurcation and it can be considered as a theoretical background for calculations presented in Kuznetsov et al. (SIAM J. Numer. Anal. 43:1407–1435, 2006). The normal form of a vector field derived in Iooss (J. Differ. Equ. 76:47–76, 1988) is not needed, since a given differential equation is considered as an algebraic equation. The theory used here concerns Fredholm operators, Lyapunov-Schmidt reduction and recognition problem for pitchfork bifurcation.

Návaznosti

EF19_073/0016943, projekt VaV

Název: Interní grantová agentura Masarykovy univerzity

MUNI/IGA/1266/2021, interní kód MU

Název: Applications of Functional Analysis and Singularity Theory to Bifurcation Theory

Investor: Masarykova univerzita, Applications of Functional Analysis and Singularity Theory to Bifurcation Theory

Citovat

ZÁTHURECKÝ, Jakub. Formal Setting for Period Doubling Bifurcation of Limit Cycles. In Skiadas, C.H., Dimotikalis, Y. 15th Chaotic Modeling and Simulation International Conference. Cham (Switzerland): Springer, 2023, s. 381-395. ISBN 978-3-031-27081-9. Dostupné z: https://doi.org/10.1007/978-3-031-27082-6_27.

@inproceedings{2297925,
   author = {Záthurecký, Jakub},
   address = {Cham (Switzerland)},
   booktitle = {15th Chaotic Modeling and Simulation International Conference},
   doi = {https://doi.org/10.1007/978-3-031-27082-6_27},
   editor = {Skiadas, C.H., Dimotikalis, Y.},
   keywords = {Limit cycle; Period doubling; Fredholm operator; Lyapunov-Schmidt reduction; Pitchfork bifurcation},
   howpublished = {tištěná verze "print"},
   language = {eng},
   location = {Cham (Switzerland)},
   isbn = {978-3-031-27081-9},
   pages = {381-395},
   publisher = {Springer},
   title = {Formal Setting for Period Doubling Bifurcation of Limit Cycles},
   url = {https://doi.org/10.1007/978-3-031-27082-6_27},
   year = {2023}
}

TY  - CONF
ID  - 2297925
AU  - Záthurecký, Jakub
PY  - 2023
TI  - Formal Setting for Period Doubling Bifurcation of Limit Cycles
PB  - Springer
CY  - Cham (Switzerland)
SN  - 9783031270819
KW  - Limit cycle
KW  - Period doubling
KW  - Fredholm operator
KW  - Lyapunov-Schmidt reduction
KW  - Pitchfork bifurcation
UR  - https://doi.org/10.1007/978-3-031-27082-6_27
N2  - A rigorous description of period doubling bifurcation of limit cycles in autonomous systems of first order differential equations based on tools of functional analysis and singularity theory is presented. It is an alternative approach which is independent of the theory of discrete-time dynamical systems, especially Poincaré sections. Particularly, sufficient conditions for its occurrence and its normal form coefficients are expressed in terms of derivatives of the operator defining given equations. Also, stability of solutions is analysed and it is related to particular derivatives of the operator. Our approach is an adjustment of techniques used by Golubitsky and Schaeffer (Singularities and Groups in Bifurcation Theory: Volume 1. Springer, New York, 1985) in the study of Hopf bifurcation and it can be considered as a theoretical background for calculations presented in Kuznetsov et al. (SIAM J. Numer. Anal. 43:1407–1435, 2006). The normal form of a vector field derived in Iooss (J. Differ. Equ. 76:47–76, 1988) is not needed, since a given differential equation is considered as an algebraic equation. The theory used here concerns Fredholm operators, Lyapunov-Schmidt reduction and recognition problem for pitchfork bifurcation.
ER  -

ZÁTHURECKÝ, Jakub. Formal Setting for Period Doubling Bifurcation of Limit Cycles. In Skiadas, C.H., Dimotikalis, Y. \textit{15th Chaotic Modeling and Simulation International Conference}. Cham (Switzerland): Springer, 2023, s.~381-395. ISBN~978-3-031-27081-9. Dostupné z: https://doi.org/10.1007/978-3-031-27082-6\_{}27.

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