Technical
CenterNational Academy of Sciences of Ukraine
Laboratory of Mathematical Modeling of Nonlinear Processes
Pokrovsʹka St., 13 Kyiv, 04070, Ukraine
Email: nonlinearlab@biomed.kiev.ua
Submitted for Publication
MixedMode Chimera States in Pendula Networks
P. Ebrahimzadeh, M. Schiek, and Y. Maistrenko
We report the emergence of peculiar chimera states in networks of identical pendula with global phaselagged coupling. The states reported include both rotating and quiescent modes, i.e. with nonzero and zero average frequencies. This kind mixedmode chimeras may be interpreted as images of bump states known in neuroscience in the context of modelling the working memory. We illustrate this striking phenomenon for a network of N = 100 coupled pendula, followed by a detailed description of the minimal nontrivial case of N = 3. Parameter regions for five characteristic types of the system behavior are identified consisting: two mixedmode chimeras with one and two rotating pendula, classical weak chimera with all three pendula rotating, synchronous rotation and quiescent state. The network dynamics is multistable: up to four of the states can coexist in the system phase state as demonstrated through the basins of attraction. The analysis suggests that the robust mixedmode chimera states can generically describe the complex dynamics of diverse pendulalike systems widespread in nature.
Recent Publications
Chimera states for directed networks.
Patrycja Jaros, Roman Levchenko, Tomasz Kapitaniak, and Yuri Maistrenko
We demonstrate that chimera behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny chimera islands arise in the parameter space. They are surrounded by developed chaotic switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, as we show for a hundred oscillators (cyclic century), the islands merge into a single chimera continent, which incorporates the world of chimeras of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as the coupling range decreases.
Generalized splay states in phase oscillator networks.
Rico Berner, Serhiy Yanchuk, Yuri Maistrenko, and Eckehard Schöll
Networks of coupled phase oscillators play an important role in the
analysis of emergent collective phenomena. In this article, we introduce
generalized 𝑚splay states constituting a special subclass of phaselocked states with vanishing 𝑚th
order parameter. Such states typically manifest incoherent dynamics,
and they often create highdimensional families of solutions (splay
manifolds). For a general class of phase oscillator networks, we provide
explicit linear stability conditions for splay states and exemplify our
results with the wellknown Kuramoto–Sakaguchi model. Importantly, our
stability conditions are expressed in terms of just a few observables
such as the order parameter or the trace of the Jacobian. As a result,
these conditions are simple and applicable to networks of arbitrary
size. We generalize our findings to phase oscillators with inertia and
adaptively coupled phase oscillator models.
Scroll ring chimera states in oscillatory networks.
Volodymyr Maistrenko, Oleksandr Sudakov, Ievgen Sliusar
We
report the appearance of a scroll ring and scroll toroid chimera states
from the proposed initial conditions for the Kuramoto model of coupled
phase oscillators in the 3D grid topology with inertia. The proposed
initial conditions provide an opportunity to obtain as single as well as
multiple scroll ring and toroid chimeras with different major and minor
diameters. We analyze their properties and demonstrate, in particular,
the patterns of coherent, partially coherent, and incoherent scroll ring
chimera states with different structures of filaments and chaotic
oscillators. Those patterns can coexist with solitary states and
solitary patterns in the oscillatory networks.
Chimera complexity.
Serhiy Brezetsky, Patrycja Jaros, Roman Levchenko, Tomasz Kapitaniak, and Yuri Maistrenko
We show an amazing complexity of the chimeras in small networks of
coupled phase oscillators with inertia. The network behavior is
characterized by heteroclinic switching between multiple saddle chimera
states and riddling basins of attractions, causing an extreme
sensitivity to initial conditions and parameters. Additional uncertainty
is induced by the presumable coexistence of stable phaselocked states
or other stable chimeras as the switching trajectories can eventually
tend to them. The system dynamics becomes hardly predictable, while its
complexity represents a challenge in the network sciences.
Solitary states in the meanfield limit.
N. Kruk, Y. Maistrenko, and H. Koeppl
We
study active matter systems where the orientational dynamics of
underlying selfpropelled particles obey second order equations. By
primarily concentrating on a spatially homogeneous setup for particle
distribution, our analysis combines theories of active matter and
oscillatory networks. For such systems, we analyze the appearance of
solitary states via a homoclinic bifurcation as a mechanism of the
frequency clustering. By introducing noise, we establish a stochastic
version of solitary states and derive the meanfield limit described by a
partial differential equation for a oneparticle probability density
function, which one might call the continuum Kuramoto model with inertia
and noise.By studying this limit, we establish second order phase
transitions between polar order and disorder. The combination of both
analytical and numerical approaches in our study demonstrates an
excellent qualitative agreement between meanfield and finite size
models.
Minimal Chimera States in PhaseLag CoupledMechanical Oscillators.
P. Ebrahimzadeh, M. Schiek, P. Jaros, T. Kapitaniak, S. van Waasen and Y. Maistrenko
We obtain experimental chimera states in the minimal network of three identical mechanical oscillators (metronomes), by introducing phaselagged alltoall coupling. For this, we have developed arealtime modelintheloop coupling mechanism that allows for flexibleand online change of coupling topology, strength and phaselag. Thechimera states manifest themselves as a mismatch of average frequencybetween two synchronous and one desynchronized oscillator. We findthis kind of striking chimeric behavior is robust in a wide parameterregion. At other parameters, however, chimera state can lose stabilityand the system behavior manifests itself as a heteroclinic switching between three saddletype chimeras. Our experimental observations arein a qualitative agreement with the model simulation.
Spiral wave chimeras for coupled oscillators with inertia.
Volodymyr Maistrenko, Oleksandr Sudakov, and Yuri Maistrenko
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, the system behavior resembles
classical twodimensional (2D) KuramotoShima spiral chimeras with
bellshape frequency characteristic of the incoherent cores. As the
coupling increases, the cores acquire concentric regions of constant
timeaveraged frequencies, the chimera becomes quasiperiodic.
Eventually, with a subsequent increase in the coupling strength, only
one such region is left, i.e., the whole core becomes
frequencycoherent. An essential modification of the system behavior
occurs, when the parameter point enters the socalled 'solitary' region.
Then, isolated oscillators are normally present on the spiral core
background of the chimera states. These solitary oscillators do not
participate in the common spiraling around the cores; instead, they
start to oscillate with different timeaveraged frequencies (Poincar\'e
winding numbers). The number and the disposition of solitary oscillators
can be any, given by the initial conditions. At a further increase in
the coupling, the spiraling disappears, and the system behavior passes
to a sort of spatiotemporal chaos.
Chimera and solitary states in 3D oscillator networks with inertia.
Volodymyr Maistrenko, Oleksandr Sudakov, and Oleksiy Osiv
Chaos 30, 063113 (2020);
https://doi.org/10.1063/5.0005281
We report the diversity of scroll wave chimeras in the
threedimensional (3D) Kuramoto model with inertia for N3 identical
phase oscillators placed in a unit 3D cube with periodic boundary
conditions. In the considered model with inertia, we have found patterns
which do not exist in a pure system without inertia. In particular, a
scroll ring chimera is obtained from random initial conditions. In
contrast to this system without inertia, where all chimera states have
incoherent inner parts, these states can have partially coherent or
fully coherent inner parts as exemplified by a scroll ring chimera.
Solitary states exist in the considered model as separate states or can
coexist with scroll wave chimeras in the oscillatory space. We also
propose a method of construction of 3D images using solitary states as
solutions of the 3D Kuramoto model with inertia.
Networkinduced multistability through lossy coupling and exotic solitary states.
Frank Hellmann, Paul Schultz, Patrycja Jaros, Roman Levchenko, Tomasz Kapitaniak, Jürgen Kurths and Yuri Maistrenko
Nature Communications 11:592 (2020); https://doi.org/10.1038/s41467020144177
The stability of synchronised networked systems is a multifaceted challenge for manynatural and technologicalfields, from cardiac and neuronal tissue pacemakers to power grids.For these, the ongoing transition to distributed renewable energy sources leads to a proliferation of dynamical actors. The desynchronisation of a few or even one of those wouldlikely result in a substantial blackout. Thus the dynamical stability of the synchronous statehas become a leading topic in power grid research. Here we uncover that, when taking intoaccount physical losses in the network, the backreaction of the network induces new exoticsolitary states in the individual actors and the stability characteristics of the synchronousstate are dramatically altered. These effects will have to be explicitly taken into account in thedesign of future power grids. We expect the results presented here to transfer to othersystems of coupled heterogeneous Newtonian oscillators.
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.
Chimera States in ThreeDimensions
Yuriy Maistrenko, Oleksandr Sudakov, Oleksiy Osiv, Volodymyr Maistrenko
New Journal of Physics, vol. 17, 073037 (2015) doi:10.1088/13672630/17/7/073037
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
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).
https://doi.org/10.1103/PhysRevE.90.032920
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
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) 10.1103/PhysRevE.89.060901
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.
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

