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
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Email: nonlinearlab@biomed.kiev.ua
Submitted for PublicationSpiral wave chimeras for coupled oscillators with inertia.
Volodymyr Maistrenko, Oleksandr Sudakov, and Yuri Maistrenko
EPJ
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, system’s behavior resembles
classical 2D Kuramoto chimeras with bellshape frequency characteristic
of the incoherent core. As the coupling increases, the core obtains
concentric regions with different constant frequencies, which means that
a chimera becomes quasiperiodic . Eventually, with a subsequent
increase in the coupling strength, only one such region is left: all
oscillators in the core become frequencycoherent. An essential
modification of system’s behavior occurs, when the parameter point
enters the socalled \solitary" parameter region, where isolated
oscillators can arise inside the spiral background. These solitary
oscillators do not participate in the common spiraling around the cores.
Instead, they start to rotate with an average frequency different from
both spiral and core ones. The disposition of solitary oscillators can
be any, as it is given by the initial conditions. At a further increase
in the coupling, the spiraling disappears, and the system passes to a
spatial or spatiotemporal chaos.
Loss of coherence among coupled oscillators: From defect states to phase turbulence.
Yohann Duguet and Yuri L. Maistrenko
Chaos 29, 121103 (2019); https://doi.org/10.1063/1.5125119
Synchronization of a large ensemble of identical phase oscillators with anonlocal kernel and a phase lag parameterαis investigated for theclassical KuramotoSakaguchi model on a ring. We demonstrate, for low enough coupling radiusrandαbelowπ/2, a phase transition betweencoherence and phase turbulence via socalleddefect states, which arise at the early stage of the transition. The defect states are a novel objectresulting from the concatenation of two or more uniformly twisted waves with different wavenumbers. Upon further increase ofα, defectslose their stability and give rise to spatiotemporal intermittency, resulting eventually in developed phase turbulence. Simulations close to thethermodynamic limit indicate that this phase transition is characterized by nonuniversal critical exponents. When many identical oscillators are coupled together, full synchronization and rotating waves are not the only possible types of dynamics. Much attention is paid to the evolution from coherent to incoherent behavior as the parameters are varied and to its universality. Here, we review one of the most studied paradigmatic models of phase oscillators coupled together in a nonlocal fashion. For this system, a nowadays classical scenario has been postulated based on the appearance of special solutions called chimera states. To our surprise, we demonstrate that for smaller coupling ranges, another route toward incoherence is the rule, based on the spatial proliferation of disorder via the interaction of defects ending eventually in spatiotemporal intermittency (STI) and phase turbulence. The present route is well described by the statistical concept of continuous phase transition and features a critical parameter below which no disorder can spread. This finding suggests new bridges between nonlinear network dynamics and statistical physics with prospective applications in various disciplines.
Dissipative solitons for bistable delayedfeedback systems.
Vladimir V. Semenov and Yuri L. Maistrenko
Chaos 28, 101103 (2018), https://doi.org/10.1063/1.5062268
We study how nonlinear delayedfeedback in the Ikeda model can induce
solitary impulses, i.e., dissipative solitons. The states are clearly
ident ified in a virtual spacetime representation of the equations with
delay, and we find that conditions for their appearance are bistability
of a nonlinear function and negative character of the delayed feedback.
Both dark and bright solitons are identified in numerical simulations
and physical electronic experiment, showing an excellent qualitative
correspondence and proving thereby the robustness of the phenomenon.
Along with single spiking solitons, a variety of compound solitonbased
structures is obtained in a wide parameter region on the route from the
regular dynamics (two quiescent states) to developed spatiotemporal
chaos. The number of coexisting solitonbased states is fast growing
with delay, which can open new perspectives in the context of
information storage. Solitons are solitary waves or wave packets travelling in space. These
structures, first reported by J. Russel in 1834, can be found in many
physical, biological, chemical, and other spatiallyextended systems.
One can distinguish solitons observed in conservative and dissipative
systems. The second ones are called dissipative solitons.
They are characterised by structural robustness and can persist for a
long time of observation despite of dissipation due to the presence of a
source of energy in an active propagation medium. Surprisingly, as it
was found in the last decade, stable localized patterns topologically
equivalent to dissipative solitons can arise in a virtual space of the
purely temporal dynamics of systems with delay. In the current paper, we
reveal the appearance of dissipative solitons in a bistable Ikedatype
system with delay. We report multiple coexistence of bright and dark
solitons from just a single one to any number as allowed by the system
size. The phenomenon is observed in a wide parameter region at the
transition from quiescence to developed spatiotemporal chaos in an
excellent qualitative correspondence between numerical simulation and
experiment. Twodimensional spatiotemporal complexity in dualdelayed nonlinear feedback systems: Chimeras and dissipative solitons.
D. Brunner, B. Penkovsky, R. Levchenko, E. Schöll, L. Larger and Y. Maistrenko
Chaos 28, 103106 (2018), https://doi.org/10.1063/1.5043391
We demonstrate for a photonic nonlinear system that two highly
asymmetric feedback delays can induce a variety of emergent patterns
which are highly robust during the system’s global evolution.
Explicitly, twodimensional chimeras and dissipative solitons become
visible upon a spacetime transformation. Switching between chimeras and
dissipative solitons requires only adjusting two system parameters,
demonstrating selforganization exclusively based on the system’s
dynamical properties. Experiments were performed using a tunable
semiconductor laser’s transmission through a FabryPérot resonator
resulting in an Airy function as nonlinearity. Resulting dynamics were
bandpass filtered and propagated along two feedback paths whose time
delays differ by two orders of magnitude. An excellent agreement between
experimental results and the theoretical model given by modified Ikeda
equations was achieved. Photonic delay systems are of astonishing diversity and have created a
rich field of fundamental research and a wide range of applications.
Under a transformation from time into pseudoscape, their basic
architecture makes them equivalent to ring networks with
perfectlysymmetric coupling. For the first time we extend this
spatiotemporal analogy in experiments by adding a second delay, 100
times the length of the first delay line. Nonlinearity is provided by a
tunable semiconductor laser traversing a FabryPérot resonator.
Visualized in 2Dspace, we show the temporal evolution of different
chimeras and dissipative solitons. Experimental results excellently
agree with numerical simulations of the doubledelay bandpass Ikeda
equation. Based on the attractors of multiple fixedpoint solutions, we
provide insight into the mechanism structuring the system’s dynamics. Selfpropelled chimeras.
Nikita Kruk,Yuri Maistrenko and Heinz Koeppl
Physical review E 98, 032219 (2018), https://doi.org/10.1103/PhysRevE.98.032219
The synchronization of selfpropelled particles (SPPs) is a fascinating instance of emergent behavior in living and manmade systems, such as colonies of bacteria, flocks of birds, robot ensembles, and many others. The recent discovery of chimera states in coupled oscillators opens up new perspectives and indicates that other emergent behaviors may exist for SPPs. Indeed, for a minimal extension of the classical Vicsek model we show the existence of chimera states for SPPs, i.e., one group of particles synchronizes while others wander aroundchaotically. Compared to chimeras in coupled oscillators where the site position is fixed, SPPs give rise to new distinctive forms of chimeric behavior. We emphasize that the found behavior is directly implied by the structure of the deterministic equation of motion and is not caused by exogenous stochastic excitation. In the scaling limit of infinitely many particles, we show that the chimeric state persists. Our findings provide the starting point forthe search or elicitation of chimeric states in real world SPP systems.
Riddling: Chimera’s dilemma.
V. Santos, J. D. Szezech, A. M. Batista, K. C. Iarosz, M. S. Baptista, H. P. Ren, C. Grebogi, R. L. Viana, I. L. Caldas, Y. L. Maistrenko, and J. Kurths
Chaos 28, 081105 (2018), https://doi.org/10.1063/1.5048595
We investigate the basin of attraction properties and its boundaries for
chimera states in a circulant network of Hénon maps. It is known that
coexisting basins of attraction lead to a hysteretic behaviour in the
diagrams of the density of states as a function of a varying parameter.
Chimera states, for which coherent and incoherent domains occur
simultaneously, emerge as a consequence of the coexistence of basin of
attractions for each state. Consequently, the distribution of chimera
states can remain invariant by a parameter change, and it can also
suffer subtle changes when one of the basins ceases to exist. A similar
phenomenon is observed when perturbations are applied in the initial
conditions. By means of the uncertainty exponent, we characterise the
basin boundaries between the coherent and chimera states, and between
the incoherent and chimera states. This way, we show that the density of
chimera states can be not only moderately sensitive but also highly
sensitive to initial conditions. This chimera’s dilemma is a consequence
of the fractal and riddled nature of the basin boundaries.
Solitary states for coupled oscillators.
Patrycja Jaros, Serhiy Brezetsky, Roman Levchenko, Dawid Dudkowski, Tomasz Kapitaniak, Yuri Maistrenko
CHAOS 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.
Chimeralike states generated by large perturbation of synchronous state of coupled metronomes.
SERGEY BREZETSKIY, DAWID DUDKOWSKI, PATRYCJA JAROS, JERZY WOJEWODA, KRZYSZTOF CZOLCZYNSKI, YURI MAISTRENKO and TOMASZ KAPITANIAK
Indian Academy of Sciences Conference Series (2017) 1:1. DOI: 10.29195/iascs.01.01.0008
Chimera
states in systems of coupled identical oscillators are spatiotemporal
patterns in which different groups of oscillators can exhibit coexisting
synchronous and incoherent behaviors despite homogeneous coupling.
Here, considering the network of coupled pendula, we find that the
patterns of chimeralike states can be generated after the large
perturbation (in which one or a few oscillators have been stopped for
the moment) of the synchronous state of the whole network. We show that
these chimeralike states can be observed in simple experiments with
mechanical oscillators, which are controlled by elementary dynamical
equations given by classical mechanics.
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

