## National Academy of Sciences of Ukraine## National Scientific Centre for Medical and Biotechnical Research## Laboratory of Mathematical Modeling of Nonlinear Processes15 Bogdana Khmelnytskogo Str, room 401 ## Chimera States in Three-DimensionsYuriy Maistrenko, Oleksandr Sudakov, Oleksiy Osiv, Volodymyr MaistrenkoNew Journal of Physics, vol. 17, 073037 (2015) doi:10.1088/1367-2630/17/7/073037 The chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. In this paper, the first evidence of three-dimensional chimera states is reported for the Kuramoto model of phase oscillators in 3D grid topology with periodic boundary conditions. Systematic analysis of the dependence of the spatiotemporal dynamics on the range and strength of coupling shows that there are two principal classes of the chimera patterns which exist in large domains of the parameter space: (I) oscillating and (II) spirally rotating. Characteristic examples from the first class include coherent as well as incoherent balls, tubes, crosses, and layers in incoherent or coherent surrounding; the second class includes scroll waves with incoherent, randomized rolls of different modality and dynamics. Numerical simulations started from various initial conditions indicate that the states are stable over the integration time. Videos of the dynamics of the chimera states are presented in supplementary material. It is concluded that three-dimensional chimera states, which are novel spatiotemporal patterns involving the coexistence of coherent and incoherent domains, can represent one of the inherent features of nature. ## Laser Chimeras as a paradigm for multi-stable patterns in complex systemsLaurent Larger, Bogdan Penkovsky, Yuri MaistrenkoNature Communications 6, Article number: 7752, doi:10.1038/ncomms8752 Chimera is a rich and fascinating class of self-organized solutions developed in high dimensional networks having non-local and symmetry breaking coupling features. Its accurate understanding is expected to bring important insight in many phenomena observed in complex spatio-temporal dynamics, from living systems, brain operation principles, and even turbulence in hydrodynamics. In this article we report on a powerful and highly controllable experiment based on optoelectronic delayed feedback applied to a wavelength tunable semiconductor laser, with which a wide variety of Chimera patterns can be accurately investigated and interpreted. We uncover a cascade of higher order Chimeras as a pattern transition from N to N-1 clusters of chaoticity. Finally, we follow visually, as the gain increases, how Chimera is gradually destroyed on the way to apparent turbulence-like system behaviour. ## Chimera states on the route from coherence to rotating wavesP. Jaros, Yu. Maistrenko, and T. KapitaniakPhysical Review E 91, 022907 (2015), doi:10.1103/PhysRevE.91.022907 We report different types of chimera states in the Kuramoto model with inertia. They arise on the route from coherence, via so-called solitary states, to the rotating waves. We identify the wide region in parameter space, in which a different type of chimera state, i.e., the imperfect chimera state, which is characterized by a certain number of oscillators that have escaped from the synchronized chimera’s cluster, appears. We describe a mechanism for the creation of chimera states via the appearance of the solitary states. Our findings reveal that imperfect chimera states represent characteristic spatiotemporal patterns at the transition from coherence to incoherence
## Different types of chimera states: An interplay between spatial and dynamical chaosDudkowski D, Maistrenko Yu., and Kapitaniak T.Physical Review E 90, 032920 (2014). We discuss the occurrence of chimera states in networks of nonlocally coupled bistable oscillators, in which individual subsystems are characterized by the coexistence of regular (a fixed point or a limit cycle) and chaotic attractors. By analyzing the dependence of the network dynamics on the range and strength of coupling, we identify parameter regions for various chimera states, which are characterized by different types of chaotic behavior at the incoherent interval. Besides previously observed chimeras with space-temporal and spatial chaos in the incoherent intervals we observe another type of chimera state in which the incoherent interval is characterized by a central interval with standard space-temporal chaos and two narrow side intervals with spatial chaos. Our findings for the maps as well as for time-continuous van der Pol–Duffing’s oscillators reveal that this type of chimera states represents characteristic spatiotemporal patterns at the transition from coherence to incoherence. ## Imperfect chimera states for coupled pendulaKapitaniak T., Kuzma P., Wojewoda J., Czolczynski K., and Maistrenko Yu.Scientific Reports, 4, 6379 (2014) The phenomenon of chimera states in the systems of coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interest. In such a state, different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Here, considering the coupled pendula, we find another pattern, the so-called imperfect chimera state, which is characterized by a certain number of oscillators which escape from the synchronized chimera’s cluster or behave differently than most of uncorrelated pendula. The escaped elements oscillate with different average frequencies (Poincare rotation number). We show that imperfect chimera can be realized in simple experiments with mechanical oscillators, namely Huygens clock. The mathematical model of our experiment shows that the observed chimera states are controlled by elementary dynamical equations derived from Newton’s laws that are ubiquitous in many physical and engineering systems. ## Cascades of Multi-headed Chimera States for Coupled Phase OscillatorsMaistrenko Yu., Vasylenko A., Sudakov O., Levchenko R., Maistrenko V. [2014]International Journal of Bifurcation and Chaos in Applied Sciences and Engineering Volume 24, Issue 08, 1440014, August 2014 http://arxiv.org/pdf/1402.1363v2.pdf DOI: 10.1142/S0218127414400148 Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of co-existing coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on
the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states
with increasing number of intervals of irregularity, so-called chimera's heads. We report three
scenarios for the chimera birth: ## Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactionsYuri Maistrenko, Bogdan Penkovsky, and Michael RosenblumPhys. Rev. E 89, 060901(R), (2014) We discuss the desynchronization transition in networks of globally coupled identical oscillators with attractive and repulsive interactions. We show that, if attractive and repulsive groups act in antiphase or close to that, a solitary state emerges with a single repulsive oscillator split up from the others fully synchronized. With further increase of the repulsing strength, the synchronized cluster becomes fuzzy and the dynamics is given by a variety of stationary states with zero common forcing. Intriguingly, solitary states represent the natural link between coherence and incoherence. The phenomenon is described analytically for phase oscillators with sine coupling and demonstrated numerically for more general amplitude models. URL: DOI: 10.1103/PhysRevE.89.060901 ## Three-Dimentional Chimera StatesVolodymyr Maistrenko, Oleksandr Sudakov, Yuri MaistrenkoEUROMECH EC565 COLLOQUIUM, 6–9 May 2014, CARGÈSE, FRANCE In this contribution, we report the first observation of three-dimensional chimera states of the following types: URL: http://perso.limsi.fr/duguet/Cargese/master.pdf |