On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators

A. S. Gonchenko; S. V. Gonchenko; A. O. Kazakov; D. V. Turaev

doi:10.1016/j.physd.2017.02.002

On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators

A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov, D. V. Turaev

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31 Scopus citations

Abstract

A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding non-conservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.

Original language	English
Pages (from-to)	45-57
Number of pages	13
Journal	Physica D: Nonlinear Phenomena
Volume	350
DOIs	https://doi.org/10.1016/j.physd.2017.02.002
State	Published - 1 Jul 2017
Externally published	Yes

Keywords

Attractor
Elliptic point
Repeller
Reversible system
Symmetry-breaking bifurcation

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

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10.1016/j.physd.2017.02.002

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title = "On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators",

abstract = "A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding non-conservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.",

keywords = "Attractor, Elliptic point, Repeller, Reversible system, Symmetry-breaking bifurcation",

author = "Gonchenko, {A. S.} and Gonchenko, {S. V.} and Kazakov, {A. O.} and Turaev, {D. V.}",

note = "Funding Information: This paper was supported by grant 14-41-00044 of the RSF . Numerical experiments in Section 5 were supported by grant RSF 14-12-00811 . The authors also acknowledge the support from the Royal Society grant IE141468 and RFBR grants 16-01-00364 and 16-51-10005 . AG was partially supported by the Program of Basic Financing by the Russian Ministry of Education and Science program no. 2000. AG and SG were partially supported by the Russian Ministry of Education and Science project 1.3287.2017 - target part. AK was supported by the Basic Research Program at the National Research University Higher School of Economics , project 90 in 2017, and by the Dynasty Foundation . DT was supported by the Weizmann Institute of Science visiting professorship program. Publisher Copyright: {\textcopyright} 2017 Elsevier B.V.",

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doi = "10.1016/j.physd.2017.02.002",

language = "English",

volume = "350",

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TY - JOUR

T1 - On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators

AU - Gonchenko, A. S.

AU - Gonchenko, S. V.

AU - Kazakov, A. O.

AU - Turaev, D. V.

N1 - Funding Information: This paper was supported by grant 14-41-00044 of the RSF . Numerical experiments in Section 5 were supported by grant RSF 14-12-00811 . The authors also acknowledge the support from the Royal Society grant IE141468 and RFBR grants 16-01-00364 and 16-51-10005 . AG was partially supported by the Program of Basic Financing by the Russian Ministry of Education and Science program no. 2000. AG and SG were partially supported by the Russian Ministry of Education and Science project 1.3287.2017 - target part. AK was supported by the Basic Research Program at the National Research University Higher School of Economics , project 90 in 2017, and by the Dynasty Foundation . DT was supported by the Weizmann Institute of Science visiting professorship program. Publisher Copyright: © 2017 Elsevier B.V.

PY - 2017/7/1

Y1 - 2017/7/1

N2 - A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding non-conservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.

AB - A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding non-conservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.

KW - Attractor

KW - Elliptic point

KW - Repeller

KW - Reversible system

KW - Symmetry-breaking bifurcation

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DO - 10.1016/j.physd.2017.02.002

M3 - Article

AN - SCOPUS:85018804909

SN - 0167-2789

VL - 350

SP - 45

EP - 57

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -

On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators

Abstract

Keywords

ASJC Scopus subject areas

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