Technical Center

National Academy of Sciences of Ukraine

Laboratory of Mathematical Modeling of Nonlinear Processes

Pokrovsʹka St., 13

Kyiv, 04070, Ukraine


Submitted Publications

Asymmetry induces critical desynchronization of power grids.    

Patrycja Jaros, Roman Levchenko, Tomasz Kapitaniak, Jurgen Kurths, and Yuri Maistrenko

Submitted to Chaos Journal

Dynamical stability of the synchronous regime remains a challenging problem for secure functioning of power grids. Based on the symmetric alternative model [Nature Communication 11, 592 (2020)], we demonstrate that the grid stability can be destroyed by elementary violations (motifs) of the network architecture such as cutting a connection between any two nodes or removing a generator or a consumer. We describe the mechanism for the cascading failure in each of the damaging case and show that the desynchronization starts with the frequency deviation of the neighbouring grid elements followed by the cascading splitting of the others, distant elements and ending eventually in the bi-modal or a partially desynchronized state. Our findings reveal that symmetric topology underlines stability of the power grids, while local damaging can cause a fatal blackout

Recent Publications

Mixed-Mode Chimera States in Pendula Networks

P. Ebrahimzadeh, M. Schiek, and Y. Maistrenko

Chaos 32, 103118 (2022),

We report the emergence of peculiar chimera states in networks of identical pendula with global phase-lagged coupling. The states reported include both rotating and quiescent modes, i.e. with non-zero and zero average frequencies. This kind mixed-mode 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 non-trivial case of N = 3. Parameter regions for five characteristic types of the system behavior are identified consisting: two mixed-mode 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 mixed-mode chimera states can generically describe the complex dynamics of diverse pendula-like systems widespread in nature.

Chimera states for directed networks.    

Patrycja Jaros, Roman Levchenko, Tomasz Kapitaniak, and Yuri Maistrenko

Chaos 31, 103111 (2021);

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

Chaos 31, 073128 (2021);

Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we introduce generalized 𝑚-splay states constituting a special subclass of phase-locked states with vanishing 𝑚th order parameter. Such states typically manifest incoherent dynamics, and they often create high-dimensional 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 well-known 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.

Illustrative video (mp4): VideoFig3a   VideoFig3b   VideoFig7

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 phase-locked 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 mean-field limit.

N. Kruk, Y. Maistrenko, and H. Koeppl

Chaos 30, 111104 (2020);

We study active matter systems where the orientational dynamics of underlying self-propelled 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 mean-field limit described by a partial differential equation for a one-particle 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 mean-field and finite size models.

Minimal Chimera States in Phase-Lag CoupledMechanical Oscillators.

P. Ebrahimzadeh, M. Schiek, P. Jaros, T. Kapitaniak, S. van Waasen and Y. Maistrenko

Eur. Phys. J. Special Topics 229, 2205 (2020).

We obtain experimental chimera states in the minimal net-work of three identical mechanical oscillators (metronomes), by intro-ducing phase-lagged all-to-all coupling. For this, we have developed areal-time model-in-the-loop coupling mechanism that allows for flexibleand online change of coupling topology, strength and phase-lag. 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 be-tween three saddle-type 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

Eur. Phys. J. Spec. Top. 229, 2327–2340 (2020).

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 two-dimensional (2D) Kuramoto-Shima spiral chimeras with bell-shape frequency characteristic of the incoherent cores. As the coupling increases, the cores acquire concentric regions of constant time-averaged 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 frequency-coherent. An essential modification of the system behavior occurs, when the parameter point enters the so-called '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 time-averaged 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);

We report the diversity of scroll wave chimeras in the three-dimensional (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.

Network-induced 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);

The stability of synchronised networked systems is a multi-faceted 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 pro-liferation 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 back-reaction 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);

Synchronization of a large ensemble of identical phase oscillators with anonlocal kernel and a phase lag parameterαis investigated for theclassical Kuramoto-Sakaguchi model on a ring. We demonstrate, for low enough coupling radiusrandαbelowπ/2, a phase transition betweencoherence and phase turbulence via so-calleddefect 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 delayed-feedback systems.

Vladimir V. Semenov and Yuri L. Maistrenko

Chaos 28, 101103 (2018),

We study how nonlinear delayed-feedback in the Ikeda model can induce solitary impulses, i.e., dissipative solitons. The states are clearly ident
ified in a virtual space-time 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 soliton-based 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 soliton-based 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 spatially-extended 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 Ikeda-type 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.

Two-dimensional spatiotemporal complexity in dual-delayed nonlinear feedback systems: Chimeras and dissipative solitons.

D. Brunner, B. Penkovsky, R. Levchenko, E. Schöll, L. Larger and Y. Maistrenko

Chaos 28, 103106 (2018),

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, two-dimensional chimeras and dissipative solitons become visible upon a space-time transformation. Switching between chimeras and dissipative solitons requires only adjusting two system parameters, demonstrating self-organization exclusively based on the system’s dynamical properties. Experiments were performed using a tunable semiconductor laser’s transmission through a Fabry-Pérot 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 pseudo-scape, their basic architecture makes them equivalent to ring networks with perfectly-symmetric 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 Fabry-Pérot resonator. Visualized in 2D-space, we show the temporal evolution of different chimeras and dissipative solitons. Experimental results excellently agree with numerical simulations of the double-delay bandpass Ikeda equation. Based on the attractors of multiple fixed-point solutions, we provide insight into the mechanism structuring the system’s dynamics.

Self-propelled chimeras.

Nikita Kruk,Yuri Maistrenko and Heinz Koeppl

Physical review E 98, 032219 (2018),

The synchronization of self-propelled 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),

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),

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, non-local, 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 saddle-type 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/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 of N3 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
oscillators, with mutual all-to-all 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 real-world systems.

Chimera-like states generated by large perturbation of synchronous state of coupled metronomes.


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 chimera-like 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 chimera-like 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 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),

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-Dimensions

Yuriy Maistrenko, Oleksandr Sudakov, Oleksiy Osiv, Volodymyr Maistrenko
New 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 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 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 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 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 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 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 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 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 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 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:
1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER,
2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and
3) via homoclinic transition with complex multistable dynamics including an "eight-like" 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.

Three-Dimentional Chimera States

Volodymyr Maistrenko, Oleksandr Sudakov, Yuri Maistrenko

In this contribution, we report the first observation of three-dimensional chimera states of the following types:
incoherent steaks, incoherent ball, and incoherent tubes (Fig.1)
to compare the phenomenon with laminar-turbulent patterns in fluid.