Abstract
Kuramoto and Battogtokh [Nonlinear Phenom. Complex Syst. 5, 380 (2002)] described chimera states as a coexistence of synchrony and asynchrony in a one-dimensional oscillatory medium. After a reformulation in terms of a local complex order parameter, the problem can be reduced to a system of partial differential equations. We further reduce finding of uniformly rotating, spatially periodic chimera patterns to solving a reversible ordinary differential equation, and demonstrate that the latter has many solutions. In the limit of neutral coupling, analytical solutions in the form of one- and two-point chimera patterns as well as localized chimera solitons are found. Based on these analytic results, patterns at weakly attracting coupling are characterized by virtue of a perturbative approach. Stability analysis reveals that only the simplest chimeras with one synchronous region are stable.
To the memory of our colleague and friend Sasha Ezersky.
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Acknowledgements
We are thankful to O. Omelchenko, M. Wolfrum, and Yu. Maistrenko for useful discussions. L. S. was supported by ITN COSMOS (funded by the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska Curie grant agreement No 642563). Stability studies were supported by the Russian Science Foundation (Project No. 14-12-00811). Poincaré section study was supported by the Russian Science Foundation (Project No. 17-12-01534).
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Smirnov, L.A., Osipov, G.V., Pikovsky, A. (2018). Chimera Patterns in One-Dimensional Oscillatory Medium. In: Abcha, N., Pelinovsky, E., Mutabazi, I. (eds) Nonlinear Waves and Pattern Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-78193-8_10
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DOI: https://doi.org/10.1007/978-3-319-78193-8_10
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