Stability of the 3-torus solution in a ring of coupled Duffing oscillators
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https://doi.org/10.1140/epjst/e2020-900276-4
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
Stability of the 3-torus solution in a ring of coupled Duffing oscillators L. Borkowski1 and A. Stefanski2,a 1
2
Department of Strength of Materials, Lodz University of Technology, ul. Stefanowskiego 1/15 Lodz, Poland Division of Dynamics, Lodz University of Technology, ul. Stefanowskiego 1/15 Lodz, Poland Received 8 December 2019 / Accepted 8 June 2020 Published online 28 September 2020 Abstract. The dynamics of the ring of unidirectionally coupled singlewell Duffing oscillators is analyzed in numerical simulation for identical nodal oscillators. The research is concentrated on the existence of the stable 3D torus attractor in this system. It is shown that 3-frequency quasi-periodicity can be robustly stable in wide range of parameters of the system under consideration. As an explanation of this stability, the conjecture on the coexistence and superposition of two independent effects characterized with irrational frequencies, i.e., the classical Newhouse, Ruelle and Takens scenario and rotating wave flow, is formulated.
1 Introduction One of typical routes to chaotic motion of nonlinear dynamical systems is a transition via quasi-periodic solutions appearing as a result of consecutive Hopf bifurcations introducing new mode with incommensurate frequency. First time such a scenario of dimensionally increasing quasiperiodicity has been proposed by Landau [1] and Hopf [2] as an explanation of transition to the turbulence. However, later work by Newhouse, Ruelle and Takens [3] had demonstrated that just after third step of this bifurcational scenario there appear chaotic strange attractor as an effect of arbitrarily small perturbation of the 3D torus (NRT scenario). On the other hand, other researchers have shown numerical [4] and experimental results that confirm the possibility of the stable 3D [5–10] or even 4D [11] torus existence. The 4D torus has been also detected for three coupled circle maps [12]. The NRT theorem has been challenged by Grebogi et al. a few years after its publication [13,14]. They have performed a numerical experiment which confirmed that smooth nonlinear perturbations do not destroy the stability of the three-frequency quasi-periodicity, what is important from physical applications point of view. The analysis of the 3D-torus stability has been continued in the last decade of the 20th century and in the current century [15–33]. Here, especially noteworthy are works by Feudel et al. [20,21] and Anischenko et al. [22]. Their justification for this phenomenon refers to the symmetry of the analyzed system. The high-dimensional quasiperiodicity (i.e. N ≥ 3) has been reported also a
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The European Physical Journal Special Topics
in sets of coupled systems [12,15–19,23–29] and self or externally driven oscillators [30–33]. Other notable cases of such solutions (four- and five-frequency torus) have been recently demonstrated in systems of chains or g
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