We study chaotic dynamics in a system of four differential equations describing the interaction of five identical phase oscillators coupled via biharmonic function. We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of a three-dimensional Poincaré map for the system under consideration. We show that chaotic dynamics develop here near a codimension three bifurcation, when a periodic orbit (fixed point of the Poincaré map) has the triplet of multipliers . As it is known, the flow normal form for such bifurcation is the well-known three-dimensional Arneodó–Coullet–Spiegel–Tresser (ACST) system, which exhibits spiral attractors. According to this, we conclude that the additional zero Lyapunov exponent for orbits in the observed attractors appears due to the fact that the corresponding three-dimensional Poincaré map is very close to the time-shift map of the ACST-system.
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September 2022
Research Article|
September 08 2022
On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators
Special Collection:
Dynamics of Oscillator Populations
Evgeny A. Grines
;
Evgeny A. Grines
a)
(Formal analysis, Investigation, Software, Visualization, Writing – original draft, Writing – review & editing)
1
Lobachevsky State University of Nizhni Novgorod
, 23 Gagarin av., Nizhny Novgorod 603950, Russia
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Alexey Kazakov
;
Alexey Kazakov
b)
(Conceptualization, Data curation, Investigation, Methodology, Project administration, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing)
2
National Research University Higher School of Economics
, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia
b)Author to whom correspondence should be addressed: kazakovdz@yandex.ru
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Igor R. Sataev
Igor R. Sataev
c)
(Data curation, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing)
3
Kotelnikov’s Institute of Radio-Engineering and Electronics of RAS
, Saratov Branch, Zelenaya 38, Saratov 410019, Russia
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a)
Electronic mail: evgenij.grines@gmail.com
b)Author to whom correspondence should be addressed: kazakovdz@yandex.ru
c)
Electronic mail: sataevir@gmail.com
Note: This article is part of the Focus Issue, Dynamics of Oscillator Populations.
Chaos 32, 093105 (2022)
Article history
Received:
May 05 2022
Accepted:
August 08 2022
Citation
Evgeny A. Grines, Alexey Kazakov, Igor R. Sataev; On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators. Chaos 1 September 2022; 32 (9): 093105. https://doi.org/10.1063/5.0098163
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