National Academy of Sciences of Ukraine
Scientific Centre for Medical and Biotechnical Research
Laboratory of Mathematical Modeling of Nonlinear Processes
15 Bogdana Khmelnytskogo Str, room 401
Kyiv, 01601, Ukraine
Phone: 380(44)2396692
Fax: 380(44)2348356
Email: nonlinearlab@biomed.kiev.ua
Recent Publications:Solitary states for coupled oscillators.
Patrycja Jaros, Serhiy Brezetsky, Roman Levchenko, Dawid Dudkowski, Tomasz Kapitaniak, Yuri Maistrenko
Chaotic 28, 011103 (2018), https://doi.org/10.1063/1.5019792
Networks of identical oscillators with inertia can display remarkable
spatiotemporal patterns in which one or a few oscillators split off from
the main synchronized cluster and oscillate with different averaged
frequency. Such “solitary states” are impossible for the classical
Kuramoto model with sinusoidal coupling. However, if inertia is
introduced, these states represent a solid part of the system dynamics,
where each solitary state is characterized by the number of isolated
oscillators and their disposition in space. We present system parameter
regions for the existence of solitary states in the case of local,
nonlocal, and global network couplings and show that they preserve in
both thermodynamic and conservative limits. We give evidence that
solitary states arise in a homoclinic bifurcation of a saddletype
synchronized state and die eventually in a crisis bifurcation after
essential variation of the parameters.
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/e2017700071
We report the appearance of threedimensional (3D) multiheaded chimera
states that display cascades of selforganized 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 headadding
cascades of the scroll wave chimeras. The phenomenon is derived for the
Kuramoto model of N^{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
socalled ‘cloning procedure’ as follows: choose a sample singleheaded
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
oscillators, with mutual alltoall coupling. Three different types of chimeras, characterized by the coexistence of
two coherent oscillators and one incoherent oscillator (i.e., rotating with another frequency) have been identified,
where the oscillators show periodic (two types) and chaotic (one type) behaviors. Typical bifurcations at the
transitions from full synchronization to chimera states and between different types of chimeras have been
described. Parameter regions for the chimera states are obtained in the form of Arnold tongues, issued from a
singular parameter point. Our analysis suggests that chimera states can be observed in small networks relevant
to various realworld systems.
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 realworld 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 socalled 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 chimeralike states are
observable in small networks relevant to various realworld systems.
Delayedfeedback chimera states: Forced multiclusters and stochastic resonance
V. Semenov, A. Zakharova, Y. Maistrenko and E. Schöll
EPL, 115 (2016) 10005, doi: 10.1209/02955075/115/10005
A nonlinear oscillator model with negative timedelayed 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 saltandpepper–like solitary
states arise on the route from regular dynamics to spatiotemporal
chaos. The control of the dynamics by external periodic forcing is
demonstrated by numerical simulations. It is shown that onecluster and
multicluster 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.
The chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a selforganized spatial pattern of coexisting coherence and incoherence. In this paper, the first evidence of threedimensional 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 threedimensional 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 multistable patterns in complex systems
Laurent Larger, Bogdan Penkovsky, Yuri Maistrenko
Nature Communications 6, Article number: 7752, doi:10.1038/ncomms8752
Chimera is a rich and fascinating class of selforganized solutions developed in high dimensional networks having nonlocal and symmetry breaking coupling features. Its accurate understanding is expected to bring important insight in many phenomena observed in complex spatiotemporal 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 N1 clusters of chaoticity. Finally, we follow visually, as the gain increases, how Chimera is gradually destroyed on the way to apparent turbulencelike system behaviour.
http://www.nature.com/ncomms/2015/150714/ncomms8752/full/ncomms8752.html
Chimera states on the route from coherence to rotating waves
P. Jaros, Yu. Maistrenko, and T. Kapitaniak
Physical 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 socalled 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 chaos
Dudkowski 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 spacetemporal 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 spacetemporal chaos and two narrow side intervals with spatial chaos. Our findings for the maps as well as for timecontinuous 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 pendula
Kapitaniak 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 socalled 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 Multiheaded Chimera States for Coupled Phase Oscillators
Maistrenko 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 selforganized spatial pattern of coexisting 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, socalled chimera's heads. We report three
scenarios for the chimera birth:
1) via saddlenode bifurcation on a resonant invariant circle, also known as SNIC or SNIPER,
2) via bluesky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddlenode periodic orbit, and
3) via homoclinic transition with complex multistable dynamics including an "eightlike" limit cycle resulting
eventually in a chimera state.
Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions
Yuri Maistrenko, Bogdan Penkovsky, and Michael Rosenblum
Phys. 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
ThreeDimentional Chimera States
Volodymyr Maistrenko, Oleksandr Sudakov, Yuri Maistrenko
EUROMECH EC565 COLLOQUIUM, 6–9 May 2014, CARGÈSE, FRANCE
In this contribution, we report the first observation of threedimensional chimera states of the following types:
incoherent steaks, incoherent ball, and incoherent tubes (Fig.1)
to compare the phenomenon with laminarturbulent patterns in fluid.
URL: http://perso.limsi.fr/duguet/Cargese/master.pdf

