Embedding nonlinear systems with two or more harmonic phase
terms near the Hopf–Hopf bifurcation

J 2023

Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation

ECLEROVÁ, Veronika, Lenka PŘIBYLOVÁ and André E. BOTHA

Basic information

Original name

Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation

Authors

ECLEROVÁ, Veronika (203 Czech Republic, guarantor, belonging to the institution), Lenka PŘIBYLOVÁ (203 Czech Republic, belonging to the institution) and André E. BOTHA

Edition

Nonlinear Dynamics, Springer Nature B.V. 2023, 0924-090X

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10102 Applied mathematics

Country of publisher

Netherlands

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 5.600 in 2022

RIV identification code

RIV/00216224:14310/23:00130030

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1007/s11071-022-07906-5

UT WoS

000863219300004

Keywords in English

Numerical continuation; Hopf–Hopf bifurcation; Neimark–Sacker bifurcation; Josephson junction; Normal form

Abstract

V originále

Nonlinear problems involving phases occur ubiquitously throughout applied mathematics andphysics, ranging from neuronal models to the search for elementary particles. The phase variables present in such models usually enter as harmonic terms and, being unbounded, pose an open challenge for studying bifurcations in these systems through standard numerical continuation techniques. Here, we propose to transform and embed the original model equations involving phases into structurally stable generalized systems that are more suitable for analysis via standard predictor–corrector numerical continuation methods. The structural stability of the generalized system is achieved by replacing each harmonic term in the original system by a supercritical Hopf bifurcation normal form subsystem. As an illustration of this general approach, specific details are provided for the ac-driven, Stewart–McCumber model of a single Josephson junction. It is found that the dynamics of the junction is underpinned by a two-parameter Hopf–Hopf bifurcation, detected in the generalized system. The Hopf–Hopf bifurcation gives birth to an invariant torus through Neimark–Sacker bifurcation of limit cycles. Continuation of the Neimark–Sacker bifurcation of limit cycles in the two-parameter space provides a complete picture of the overlapping Arnold tongues (regions of frequency-locked periodic solutions), which are in precise agreement with the widths of the Shapiro steps that can be measured along the current–voltage characteristics of the junction at various fixed values of the ac-drive amplitude.

Links

EF16_013/0001761, research and development project

Name: RECETOX RI

EF17_043/0009632, research and development project

Name: CETOCOEN Excellence

MUNI/A/1342/2021, interní kód MU

Name: Matematické a statistické modelování 6 (Acronym: MaStaMo6)

Investor: Masaryk University

MUNI/A/1615/2020, interní kód MU

Name: Matematické a statistické modelování 5 (Acronym: MaStaMo5)

Investor: Masaryk University

90121, large research infrastructures

Name: RECETOX RI

Citovat

ECLEROVÁ, Veronika, Lenka PŘIBYLOVÁ and André E. BOTHA. Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation. Nonlinear Dynamics. Springer Nature B.V., 2023, vol. 111, No 2, p. 1537-1551. ISSN 0924-090X. Available from: https://dx.doi.org/10.1007/s11071-022-07906-5.

@article{2221489,
   author = {Eclerová, Veronika and Přibylová, Lenka and Botha, André E.},
   article_number = {2},
   doi = {http://dx.doi.org/10.1007/s11071-022-07906-5},
   keywords = {Numerical continuation; Hopf–Hopf bifurcation; Neimark–Sacker bifurcation; Josephson junction; Normal form},
   language = {eng},
   issn = {0924-090X},
   journal = {Nonlinear Dynamics},
   title = {Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation},
   url = {https://link.springer.com/article/10.1007/s11071-022-07906-5},
   volume = {111},
   year = {2023}
}

TY  - JOUR
ID  - 2221489
AU  - Eclerová, Veronika - Přibylová, Lenka - Botha, André E.
PY  - 2023
TI  - Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation
JF  - Nonlinear Dynamics
VL  - 111
IS  - 2
SP  - 1537-1551
EP  - 1537-1551
PB  - Springer Nature B.V.
SN  - 0924090X
KW  - Numerical continuation
KW  - Hopf–Hopf bifurcation
KW  - Neimark–Sacker bifurcation
KW  - Josephson junction
KW  - Normal form
UR  - https://link.springer.com/article/10.1007/s11071-022-07906-5
N2  - Nonlinear problems involving phases occur ubiquitously throughout applied mathematics andphysics, ranging from neuronal models to the search for elementary particles. The phase variables present in such models usually enter as harmonic terms and, being unbounded, pose an open challenge for studying bifurcations in these systems through standard numerical continuation techniques. Here, we propose to transform and embed the original model equations involving phases into structurally stable generalized systems that are more suitable for analysis via standard predictor–corrector numerical continuation methods. The structural stability of the generalized system is achieved by replacing each harmonic term in the original system by a supercritical Hopf bifurcation normal form subsystem. As an illustration of this general approach, specific details are provided for the ac-driven, Stewart–McCumber model of a single Josephson junction. It is found that the dynamics of the junction is underpinned by a two-parameter Hopf–Hopf bifurcation, detected in the generalized system. The Hopf–Hopf bifurcation gives birth to an invariant torus through Neimark–Sacker bifurcation of limit cycles. Continuation of the Neimark–Sacker bifurcation of limit cycles in the two-parameter space provides a complete picture of the overlapping Arnold tongues (regions of frequency-locked periodic solutions), which are in precise agreement with the widths of the Shapiro steps that can be measured along the current–voltage characteristics of the junction at various fixed values of the ac-drive amplitude.
ER  -

ECLEROVÁ, Veronika, Lenka PŘIBYLOVÁ and André E. BOTHA. Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation. \textit{Nonlinear Dynamics}. Springer Nature B.V., 2023, vol.~111, No~2, p.~1537-1551. ISSN~0924-090X. Available from: https://dx.doi.org/10.1007/s11071-022-07906-5.

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