## National Academy of Sciences of Ukraine## Scientific Centre for Medical and Biotechnical Research## Laboratory of Mathematical Modeling of Nonlinear Processes15 Bogdana Khmelnytskogo Str, room 401 ## Submitted Papers:## Solitary states for coupled oscillators.Patrycja Jaros, Serhiy Brezetsky, Roman Levchenko, Dawid Dudkowski, Tomasz Kapitaniak, Yuri Maistrenko Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS), arXiv:1703.06950 ## We demonstrate that solitary states can be widely observed for networks of coupled oscillators with local, non-local and global couplings, and they preserve in both thermodynamic and Hamiltonian limits. We show that depending on units' and network's parameters, different types of solitary states occur, characterized by the number of isolated oscillators and the disposition in space. The creation of solitary states through the homoclinic bifurcation is described and the regions of co-existence of obtained states and typical examples of dynamics have been identified. Our analysis suggests that solitary states can be observed in a wide class of networks relevant to various real-world systems.## Recent Publications:## Multiple scroll wave chimera states.Volodymyr Maistrenko, Oleksandr Sudakov, Oleksiy Osiv and Yuri Maistrenko The European Physical Journal Special Topics, June 2017, Volume 226, Issue 9, pp 1867–1881, DOI 10.1140/epjst/e2017-70007-1 We report the appearance of three-dimensional (3D) multiheaded chimera states that display cascades of self-organized spatiotemporal patterns of coexisting coherence and incoherence. We demonstrate that the number of incoherent chimera domains can grow additively under appropriate variations of the system parameters generating thereby head-adding cascades of the scroll wave chimeras. The phenomenon is derived for the Kuramoto model ofN^{3} identical phase oscillators placed in the unit 3D cube with periodic boundary conditions, parameters being the coupling radius r
and phase lag α. To obtain the multiheaded chimeras, we perform the
so-called ‘cloning procedure’ as follows: choose a sample single-headed
3D chimera state, make appropriate scale transformation, and put some
number of copies of them into the unit cube. After that, start numerical
simulations with slightly perturbed initial conditions and continue
them for a sufficiently long time to confirm or reject the state
existence and stability. In this way it is found, that multiple scroll
wave chimeras including those with incoherent rolls, Hopf links and
trefoil knots admit this sort of multiheaded regeneration. On the other
hand, multiple 3D chimeras without spiral rotations, like coherent and
incoherent balls, tubes, crosses, and layers appear to be unstable and
are destroyed rather fast even for arbitrarily small initial
perturbations.
## Smallest chimera states.Yuri Maistrenko, Serhiy Brezetsky, Patrycja Jaros, Roman Levchenko and Tomasz Kapitaniak. PHYSICAL REVIEW E 95, 010203(R) (2017). doi: 10.1103/PhysRevE.95.010203. We demonstrate that chimera behavior can be observed in small networks consisting of three identical ## The smallest chimera state for coupled pendula.Wojewoda J, Czolczynski K, Maistrenko Y, Kapitaniak T. Sci Rep. 2016 Oct 7;6:34329. doi: 10.1038/srep34329. Chimera states in the systems of coupled identical oscillators are spatiotemporal patterns in which different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Although these states are typically observed in large ensembles of oscillators, recently it has been suggested that chimera states may occur in the systems with small numbers of oscillators. Here, considering three coupled pendula showing chaotic behavior, we find the pattern of the smallest chimera state, which is characterized by the coexistence of two synchronized and one incoherent oscillator. We show that this chimera state can be observed in simple experiments with mechanical oscillators, which are controlled by elementary dynamical equations derived from Newton's laws. Our finding suggests that chimera states are observable in small networks relevant to various real-world systems. ## Occurrence and stability of chimera states in coupled externally excited oscillators.Dawid Dudkowski, Yuri Maistrenko and Tomasz Kapitaniak. Chaos 26, 116306 (2016), http://dx.doi.org/10.1063/1.4967386 We studied the phenomenon of chimera states in networks of non–locally coupled externally excited oscillators. Units of the considered networks are bi–stable, having two co–existing attractors of different types (chaotic and periodic). The occurrence of chimeras is discussed, and the influence of coupling radius and coupling strength on their co–existence is analyzed (including typical bifurcation scenarios). We present a statistical analysis and investigate sensitivity of the probability of observing chimeras to the initial conditions and parameter values. Due to the fact that each unit of the considered networks is individually excited, we study the influence of the excitation failure on stability of observed states. Typical transitions are shown, and changes in network's dynamics are discussed. We analyze systems of coupled van der Pol–Duffing oscillators and the Duffing ones. Described chimera states are robust as they are observed in the wide regions of parameter values, as well as in other networks of coupled forced oscillators.
## Experimental multistable states for small network of coupled pendula.Dawid Dudkowski, Juliusz Grabski, Jerzy Wojewoda, Przemyslaw Perlikowski, Yuri Maistrenko and Tomasz Kapitaniak. Scientific reports 6, 29833 (2016), doi:10.1038/srep29833 Chimera states are dynamical patterns emerging in populations of coupled identical oscillators where different groups of oscillators exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Although these states are typically observed in the large ensembles of oscillators, recently it has been shown that so-called weak chimera states may occur in the systems with small numbers of oscillators. Here, we show that similar multistable states demonstrating partial frequency synchronization, can be observed in simple experiments with identical mechanical oscillators, namely pendula. The mathematical model of our experiment shows that the observed multistable states are controlled by elementary dynamical equations, derived from Newton’s laws that are ubiquitous in many physical and engineering systems. Our finding suggests that multistable chimera-like states are observable in small networks relevant to various real-world systems.
Delayed-feedback chimera states: Forced multiclusters and stochastic resonance
V. Semenov, A. Zakharova, Y. Maistrenko and E. Schöll EPL, 115 (2016) 10005, doi: 10.1209/0295-5075/115/10005## A nonlinear oscillator model with negative time-delayed feedback is studied numerically under external deterministic and stochastic forcing. It is found that in the unforced system complex partial synchronization patterns like chimera states as well as salt-and-pepper–like solitary states arise on the route from regular dynamics to spatio-temporal chaos. The control of the dynamics by external periodic forcing is demonstrated by numerical simulations. It is shown that one-cluster and multi-cluster chimeras can be achieved by adjusting the external forcing frequency to appropriate resonance conditions. If a stochastic component is superimposed to the deterministic external forcing, chimera states can be induced in a way similar to stochastic resonance, they appear, therefore, in regimes where they do not exist without noise.## 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 |