Spiral wave chimeras for coupled oscillators with inertia
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part of Springer Nature, 2020 https://doi.org/10.1140/epjst/e2020-900279-x
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
Spiral wave chimeras for coupled oscillators with inertia Volodymyr Maistrenko1,a , Oleksandr Sudakov1,2 , and Yuri Maistrenko1,3 1
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Scientific Center for Medical and Biotechnical Research, NAS of Ukraine, Volodymyrska Str. 54, 01030 Kyiv, Ukraine Taras Shevchenko National University of Kyiv, Volodymyrska Str. 60, 01030 Kyiv, Ukraine Institute of Mathematics, NAS of Ukraine, Tereshchenkivska Str. 3, 01024 Kyiv, Ukraine Received 12 December 2019 / Accepted 8 June 2020 Published online 28 September 2020 Abstract. We report the appearance and the metamorphoses of spiral wave chimera states in coupled phase oscillators with inertia. First, when the coupling strength is small enough, the system behavior resembles classical two-dimensional (2D) Kuramoto-Shima spiral chimeras with bell-shape frequency characteristic of the incoherent cores [Y. Kuramoto, S.I. Shima, Prog. Theor. Phys. Supp. 150, 115 (2003); S.I. Shima, Y. Kuramoto, Phys. Rev. E. 69, 036213 (2004)]. As the coupling increases, the cores acquire concentric regions of constant time-averaged frequencies, the chimera becomes quasiperiodic. Eventually, with a subsequent increase in the coupling strength, only one such region is left, i.e., the whole core becomes frequency-coherent. An essential modification of the system behavior occurs, when the parameter point enters the so-called solitary region. Then, isolated oscillators are normally present on the spiral core background of the chimera states. These solitary oscillators do not participate in the common spiraling around the cores; instead, they start to oscillate with different timeaveraged frequencies (Poincar´e winding numbers). The number and the disposition of solitary oscillators can be any, given by the initial conditions. At a further increase in the coupling, the spiraling disappears, and the system behavior passes to a sort of spatiotemporal chaos.
1 Introduction Spiral wave chimeras are fascinating two-dimensional (2D) patterns first reported in 2003 by Kuramoto and Shima [1,2]. Manifesting the regular 2D spiraling, this kind of patterns possesses, nevertheless, finite-sized incoherent cores with a bell-shaped frequency distribution of individual oscillators. Since that, spiral wave chimeras have been intensively studied in the Kuramoto-Sakaguchi and other models, both numerically and analytically [3–13], and they got recently an experimental confirmation in [11]. a
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The European Physical Journal Special Topics
In this paper, with the use of a detailed numerical study, we demonstrate the appearance of spiral wave chimeras for Kuramoto model with inertia, and we follow their transformations with increasing the coupling strength µ. We begin with a standard chimeric pattern including a number of incoherent bell-shaped cores and then go to the situation where the cores become coherent. The latter means that all in-core oscill
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