Chimera States: From Theory and Experiments to Technology and Living Systems

Coherence and randomness coexist in light sources and influence their applications, such as optical communications and imaging. Measures of spatio-temporal coherence of light are traditionally provided by correlation functions. As a laser makes the transition from below threshold, where it is dominated by spontaneous emission and incoherence, to operation above threshold where stimulated emission dominates, passage time statistics can be used to quantify the simultaneous coexistence of coherence and incoherence that leads to spontaneous switching of lasing modes. With the recent exploration of chimera states in networks of coupled oscillators has emerged the possibility of noise initiated switching between two sub-chimeras. We will describe simulations that revealed power-law switching between the sub-chimeras and led to experiments on coupled opto-electronic oscillators to measure passage time statistics for switching chimeras.

Si electrodissolution in fluoride-containing electrolytes exhibits a multitude of self-organized spatio-temporal states, among them chimera states and other coexistence patterns as well as multi-frequency clusters. In this talk, I will discuss that most of these patterns emerge due to nonlinear coupling, which might be global or nonlocal depending on the parameter ranges. Model caclulations with an ensemble of globally coupled Stuart-Landau oscillators with nonlinear coupling show that such an ensemble can express a range of coesxistence patterns more comprehensive than chimeras.(1) A hierarchy of coexisting states evolves from the fully synchronized solution in a series of cluster-splittings. At the far end of this hierarchy, the states further collide with their own mirror-images in phase space – rendering the motion chaotic, destroying some of the clusters and thereby producing even more intricate coexistence patterns. A sequence of such attractor collisions can ultimately lead to full incoherence of only single asynchronous oscillators. Chimera states, with one large synchronized cluster and else only single oscillators, are found to be just one step in this transition from low- to high-dimensional dynamics. On the other hand, an analysis of the experimental conditions where multi-frequency clusters were observed, suggests that the spatial coupling in this parameter range is nonlinear and nonlocal, with the strength of the coupling adapting to the state of the oscillators, similar to what is known from the context of neural dynamics.(2) We will critically discuss the relationship between multifrequency clusters and chimeras. (1) S.W. Haugland, A. Tosolini, and K. Krischer, Nature Communications (2021) 12, 5634. (2) M.Patzauer and K. Krischer, Phys. Rev. Lett. (2021) 126, 194101

A remarkable quantum scattering phenomenon in two-dimensional Dirac material systems will be presented, where the manifestations of both classically integrable and chaotic dynamics emerge simultaneously and are electrically controllable. The distinct relativistic quantum fingerprints associated with different electron spin states are due to a physical mechanism analogous to chiroptical effect in the presence of degeneracy breaking. The phenomenon mimics a chimera state in classical complex dynamical systems but here in a relativistic quantum setting - henceforth the term "Dirac quantum chimera," associated with which are physical phenomena with potentially significant applications such as enhancement of spin polarization, unusual coexisting quasibound states for distinct spin configurations, and spin selective caustics. Experimental observations of these phenomena are possible through, e.g., optical realizations of ballistic Dirac fermion systems.

Experimental progress in optomechanical systems, in trapped-ion setups, and in superconducting circuit-QED architectures has motivated the study of synchronization in quantum systems. In my talk I would like to describe theoretical approaches to the synchronization problem for quantum oscillators and discuss some of the issues and open questions [1-4]. Presently, in collaboration with Martin Koppenhöfer (University of Chicago) and Tobias Nadolny (University of Basel), we are studying small networks of quantum oscillators with the goal to find quantum chimera states. [1] N. Lörch, S.E. Nigg, A. Nunnenkamp, R.P. Tiwari, and C. Bruder, Phys. Rev. Lett. 118, 243602 (2017). [2] A. Roulet and C. Bruder, Phys. Rev. Lett. 121, 053601 (2018). [3] A. Roulet and C. Bruder, Phys. Rev. Lett. 121, 063601 (2018). [4] M. Koppenhöfer, C. Bruder, and A. Roulet, Phys. Rev. Research 2, 023026 (2020).

The human brain is a complex dynamical system where patterns of neurophysiological activity evolve rapidly, forming transient domains of synchronization across subsets of brain regions. How cognition emerges from these spatiotemporal patterns of regional brain activity remains an open question. Due to their natural ability to describe partial synchrony, chimera states have an intuitive utility for augmenting our understanding of the brain’s function. I will discuss the application of a chimera-based framework in understanding how different groups of brain regions interconnect to form diverse patterns of synchronization. In particular, I will talk about brain network models driven by the actual anatomical connectivity of the human brain and emergent patterns of chimera states in these models.

Neural synchrony in the brain is usually variable and intermittent even at rest, thus intervals of predominantly synchronized activity are interrupted by intervals of desynchronized activity. While the mechanisms of this kind of dynamics are not fully explored, we have developed the time-series analysis techniques to characterize the temporal variability of synchrony in neurophysiological experimental data and found the correlations between the temporal patterning of neural synchrony and several behavioral effects. I will present our studies of experimental data in Parkinson’s disease, disorders of addiction, and autism spectrum disorders. I will further discuss potential mechanisms behind this observed synchrony variability.

Synchronized and partially synchronized brain activity plays a key role in normal cognition and in some neurological disorders, such as epilepsy. However, the mechanism by which synchrony and asynchrony co-exist in a population of neurons remains elusive. Chimera states, where synchrony and asynchrony coexist, have been documented only for precisely specified connectivity and network topologies. Here, we demonstrate how Chimeras can emerge in Recurrent Neural Networks (RNN) by training the networks to display Chimeras with machine learning. These solutions, which we refer to as embedded Chimeras, are generically produced by RNNs with connectivity matrices only slightly perturbed from random networks. We also demonstrate that learning is robust to different biological constraints, such as the excitatory/inhibitory classification of neurons (Dale's law), and the sparsity of connections in neural circuits. The RNN can also be trained to switch Chimera solutions: an input pulse can trigger the RNN to switch the synchronized and the unsynchronized groups of the embedded Chimera, reminiscent of uni-hemispheric sleep in a variety of animals. Our results imply that the emergence of Chimeras is quite generic at the meso- and macroscale suggesting their general relevance in neuroscience.

The study of the collective behavior and emergent dynamics in nonlinear oscillator networks has applications throughout physics, biology, and computational neuroscience. In this context, the Kuramoto Model (KM) plays an important role in the modeling and analysis of these systems. However, the precise relationship between the structure of a specific network and the emergence of complex spatiotemporal patterns remains unclear. To address this issue, we present a new mathematical approach that allows us to directly relate the connectivity of an individual network with the spatiotemporal pattern of oscillations and the collective behavior of the system. To do this, we use a complex-valued matrix formulation for the KM, whose argument has a correspondence with the original model. This allows an analytical approach to study three major synchronization phenomena (phase synchronization, chimera states, and traveling waves) in oscillator networks for finite time and in individual realizations. Moreover, it reveals a possible underlying mechanism for the emergence of these phenomena by connecting the spectrum of the matrix describing the system with the resulting dynamical behavior. This approach gives us a new, geometric perspective of synchronization phenomena in terms of complex eigenmodes of the system, which in turn offers a unified geometry for synchrony, chimera states, and waves in nonlinear oscillator networks. Finally, it opens a new opportunity to better understand the relation between network structure and complex dynamics, such as chimera states, in the context of neural systems and computation, where the Kuramoto model has been successfully applied.

Rhythms influence our life in various ways, e.g., through heart beat and respiration, oscillating brain currents, life cycles and seasons, clocks and metronomes, pulsating lasers, transmission of data packets, and many others. The physics of complex nonlinear systems has developed methods to describe and analyze self-sustained oscillations and their synchronization in complex networks, which are composed of many components. Synchronized oscillations as well as completely asynchronous chaotic oscillations play a major role in many networks in nature and technology. For instance, the synchronous firing of all neurons in the brain represents a pathological state, like in epilepsy or Parkinson's disease, and should be suppressed, as well as the synchronous mechanical vibration of bridges. On the other hand, synchronization is desirable for the stable operation of power grids or in encrypted communication with chaotic signals. In networks composed of identical components, intriguing hybrid states ("chimeras") may form spontaneously, which consist of spatially coexisting synchronized and desynchronized domains, i.e., seemingly incongruous parts. This might be of relevance in inducing and terminating epileptic seizures, or in unihemispheric sleep which is found in certain migratory birds and mammals, or in cascading failures of the power grid.

Multilayer networks offer a better representation of the topology and dynamics of real-world systems in comparison with isolated one-layer structures. The prime objective of multilayer networks is to explore multiple levels of interactions where functions of one layer get affected by the properties of other layers. One of the most promising applications of the multilayer approach is the study of the brain, or technological interdependent systems, i.e., those systems in which the correct functioning of one of them strongly depends on the status of the others. Multiplexing plays an important role in controlling since it is not always possible to directly access and manipulate the desired layer, while the network it is multiplexed with may be adaptable. We investigate multilayer networks of coupled FitzHugh-Nagumo neurons and focus on the case of weak multiplexing, i.e., when the coupling between the layers is smaller than that inside the layers. It turns out that weak multiplexing has an essential impact on the dynamical patterns observed in the system and can be used for controlling in both oscillatory and excitable regimes. In the oscillatory regime, we show that different types of partial synchronization patterns such as chimera states [1-2] and solitary states [3-4] can be induced and suppressed. In the excitable regime with noise, we find that weak multiplexing induces coherence resonance in networks that do not demonstrate this phenomenon in isolation [5]. [1] A. Zakharova, Chimera Patterns in Networks: Interplay between Dynamics, Structure, Noise, and Delay, Understanding Complex Systems, Springer, ISBN 978-3-030-21713-6, (2020) [2] M. Mikhaylenko, L. Ramlow, S. Jalan, and A. Zakharova, Weak multiplexing in neural networks: Switching between chimera and solitary states, Chaos 29, 023122 (2019) [3] E. Rybalova, V. S. Anishchenko, G. I. Strelkova, A. Zakharova, Solitary states and solitary state chimera in neural networks, Chaos Fast Track 29, 071106 (2019) [4] L. Schülen, S. Ghosh, A. D. Kachhvah, A. Zakharova, S. Jalan, Delay engineered solitary states in complex networks, Chaos, Solitons and Fractals 128, 290 (2019) [5] N. Semenova, A. Zakharova, Weak multiplexing induces coherence resonance, Chaos 28, 051104 (2018) *Selected as Editor’s Pick

Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and oscillates with a different frequency. Such chimera-type patterns with an incoherent state formed by a single oscillator were observed in various oscillator networks; however, there is still a lack of understanding of how such states can stably appear. Here, we study the stability of solitary states in Kuramoto networks of identical two-dimensional phase oscillators with inertia and a phase-lagged coupling. The presence of inertia can induce rotatory dynamics of the phase difference between the solitary oscillator and the coherent cluster. We derive asymptotic stability conditions for such a solitary state as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (1) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (2) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our stability analysis for the emergence of rotatory chimeras. This is a joint work with V. Munyayev, M. Bolotov , L. Smirnov, and G. Osipov.

Chimera states, characterized by coexisting regions of phase coherence and incoherence, are stable patterns even in a system with translational and rotational symmetry. Its existence has been experimentally demonstrated in chemical, mechanical, electronic, and opto-electronic systems. Here, we show the formation of chimera patterns should also be observable in ultracold atoms. More generally, chimera patterns exists in conservative Hamiltonian systems with long-range nonlocal hopping in which both energy and particle number are conserved. In particular, we outline the implementation in a two-component Bose-Einstein condensate with a spin-dependent optical lattice. The calculated parameter regime are well within the current implementable technology.

Chimera states have attracted significant attention as symmetry-broken states exhibiting the coexistence of coherence and incoherence. Despite the valuable insights gained by analyzing specific systems, the understanding of the physical mechanism underlying the emergence of chimeras has been incomplete. In this presentation, I will argue that an important class of stable chimeras arise because coherence in part of the system is sustained by incoherence in the rest of the system. This mechanism may be regarded as a deterministic analog of noise-induced synchronization and is shown to underlie the emergence of so-called strong chimeras. These are chimera states whose coherent domain is formed by identically synchronized oscillators. The link between coherence and incoherence revealed by this mechanism also offers insights into the dynamics of a broader class of states, including switching chimera states. References: Y. Zhang and A. E. Motter, Mechanism for strong chimeras, Phys. Rev. Lett. 126, 094101 (2021); Y. Zhang, Z. G. Nicolaou, J. D. Hart, R. Roy, and A. E. Motter, Critical switching in globally attractive chimeras, Phys. Rev. X 10, 011044 (2020).

Systems of nonlocally coupled oscillators have been the subject of much research over the past decades. It has been found that in many cases they support complex spatiotemporal patterns of coexisting coherence and incoherence, called chimera states. Until now, such chimera states have been studied mostly in the stationary case, when the shape and position of the chimera pattern remain unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing or moving chimera states, were also reported, however not investigated systematically because of their complexity. In this talk, we report a number of theoretical results related to the latter nonstationary patterns. In particular, we show typical bifurcation diagrams that mediate their appearance and reveal the constructive role of oscillator heterogeneities for the stability of these patterns.

Due to heterogeneity Kuramoto models exhibit frustrated synchronization around the synchronization transition. Power-grids can be described by the second order Kuramoto model. I provide numerical evidence for an extended coupling region, where blackout cascade size distribution exhibit exhibit power-law tails and show the spatial structure of the local Kuramoto order parameter [1,2] For modelling the brain the first order Kuramoto serves as a simplest oscillatory model. On heterogenous, modular connectome graphs we can find extended dynamical critical region with large steady state fluctuations as compared to random graphs [3,4,5]. We show new results for the local Kuramto maps on connectomes. [1] Géza Ódor and Bálint Hartmann, Heterogeneity effects in power grid network models, Phys. Rev. E 98 (2018) 022305 [2] Géza Ódor and Bálint Hartmann, Power-Law Distributions of Dynamic Cascade Failures in Power-Grid Models Entropy 22 (2020) 666 [3] Géza Ódor and Jeffrey Kelling, Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs, Scientific Reports 9 (2019) 19621 [4] Géza Ódor, Jeffrey Kelling and Gustavo Deco, The effect of noise on the synchronization dynamics of the Kuramoto model on a large human connectome graph Neurocomputing, 461 (2021) 696-704. [5] Géza Ódor, Gustavo Deco and Jeffrey Kelling, What makes us humans: Differences in the critical dynamics underlying the human and fruit-fly connectome arXiv:2201.11084, acceptrd in Phys. Rev. Res.

Recently, a new phenomenon, called solitary state, in the transition from incoherence to complete coherence was discovered. We analyze networks of adaptively coupled phase oscillators and observe a variety of frequency-synchronized states including solitary states. Despite the fact that solitary states have been observed in a plethora of dynamical systems, the mechanisms behind their emergence were largely unaddressed in the literature. Here, we show how solitary states and other partial synchronization patterns emerge due to the adaptive feature of the network. By using numerical and analytical methods, we classify several bifurcation scenarios in which these states are created and stabilized.

In everyday speech the word ‘chimera’ is used to describe objects composed of inharmonious parts. In the field of dynamical systems there exist a term "weak chimera state", which is defined by at least one oscillator rotating with a different average frequency in comparison to at least two others, which are frequency synchronized. When coupling small number of phase oscillators with inertia, weak chimera states can be obtained. They are present in the form of in-phase, anti-phase or rotating wave chimeras. Each of these chimeras may occur as one of its permutations. Parameter regions of the stable chimera states are surrounded by the so-called riddling shadow, where the network behavior represents heteroclinic switching between multiple saddle chimera states (i.e. permutationally different chimeras). Basins of the chimeras are riddled, what cause extreme sensitivity of the switching solutions to initial conditions and the parameters. Another strange puzzle is induced by a presumable presence of antipodal points or other chimera states as the switching trajectories then can tend to one of them. The dynamics becomes hardly predictable.

The co-existence of multistable rhythms generated by oscillatory neural circuits made up of 4 and more cells, their onset, stability conditions, and the transitions between such rhythms are not well understood. This is partly due to the lack of appropriate visual and computational tools. In this study, we employ modern computational approaches including unsupervised machine learning (clustering) algorithms and fast parallel simulations powered by graphics processing units (GPUs) to further extend our previously developed techniques based on the theory of dynamical systems and bifurcations. This allows us to analyze the fundamental principles and mechanisms that ensure the robustness and multifunctionality of such neural circuits. In addition, we examine how network topology affects the dynamics, and the rhythmic patterns transition/bifurcate as network configurations are altered and the intrinsic properties of the cells and the synapses are varied. This study elaborates on a set of inhibitory coupled 4-cell circuits that can exhibit a variety of mono- and multistable rhythms including pacemakers, paired half-centers, traveling- waves, synchronized states, as well as various chimera states which are characterized by two sub-populations firing at distinct frequencies.. Our detailed analysis is helpful to generate verifiable hypotheses for neurophysiological experiments with biological central pattern generators. Krishna Pusuluri, Sunitha Basodi, Andrey Shilnikov Neuroscience Institute Georgia State University USA

We report the emergence of chimera states as frequency clusters in the network of identical pendulua with phase-lag coupling. Such clusters include multiple rotation mode pendulums, where average frequencies are non-zero, and an oscillation mode cluster with zero average frequency. The dynamics of each cluster lies on a synchronization manifold with mixed mode oscillations. The formation of mix mode chimeras are studied in detail for the minimal network of only $N=3$ pendulums. Parameter region for the coexistence of different combinations of the mix mode chimera states are obtained and the dependence on the initial conditions are shown through the multiple basins of attraction. The analysis suggests that the robust mixed mode chimera states can commonly characterize the complex dynamics of diverse pendula-like systems widespread in the nature.

We discuss the analytical and numerical approaches to phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter. Particularly, for three coupled Stuart–Landau (SL) oscillators, where the phase can be introduced explicitly, an analytic perturbation procedure yields the explicit second-order approximation [1]. We exploit the analytical result from [1] to analyze the mechanism of the remote synchronization (RS). RS, briefly reported by Okuda and Kuramoto as early as 1991, implies that oscillators interacting not directly but via an additional unit (hub) adjust their frequencies and exhibit frequency locking while the hub remains asynchronous. Previous studies uncovered the role of amplitude dynamics and of nonisochronicity: RS appeared in a network of isochronous SL units but not in its first-order phase approximation, the Kuramoto network. Furthermore, RS emerged in networks of phase oscillators with the Kuramoto-Sakaguchi interaction, but not in the case of zero phase shift in the sine-coupling term; this result indicates the role of nonisochronicity. In this work, we analytically demonstrate the role of two factors promoting remote synchrony. These factors are the nonisochronicity of oscillators and the coupling terms appearing in the second-order phase approximation. We explain the contribution of both factors and quantitatively describe the transition to RS. We demonstrated that the RS transition is determined by the interplay of the nonisochronicity and the amplitude dynamics. The impact of the latter factor renders the standard first-order phase dynamics descripion of the RS phenomenon invalid. Our result emphasizes the importance of higher-order phase reduction and highlights the crucial role amplitude dynamics may have in governing the behavior of networks of nonlinear oscillators. We show a good correspondence between our theory and numerical results for small and moderate coupling strengths and argue that the effect of the amplitude dynamics neglected in the first-order phase approximation and revealed by the higher-order one holds for general limit-cycle oscillators. [1] E. Gengel, E. Teichmann, M. Rosenblum, and A. Pikovsky, J. of Physics: Complexity, 2, 015005 (2021) [2] M. Kumar and M. Rosenblum, PRE 104, 054202 (2021)

When electrochemical reactions take place on electrode arrays, a network can form through the potential drop among the elements. In the presentation, experimental results are shown that confirm the existence of weak chimera states in modular networks of the nickel electrodissolution system with two approaches. In the first approach, we consider oscillatory dynamics close to a Hopf bifurcation. We employ synchronization engineering to design nonlinear delayed feedback to induce a weak chimera state, where one group of two oscillators are in-phase, and another group of oscillators are anti-phase; the elements in the same groups have the same, but the elements in the different groups have different frequencies. Thus, in this example, weakly nonlinear oscillations generate chimera state with strongly nonlinear coupling. In the second approach, we consider moderately relaxation oscillators. The parameters that favor chimera states are found with experiments with globally coupled oscillators in the form of co-existing one- and two-cluster states. Experiments with two groups oscillator confirm the weak chimera state, where one group is in one-cluster, and the other group is in a two-cluster state; the one and the two-cluster states have different frequency. Thus, in this second example, strongly nonlinear oscillators generate chimera state with linear (difference) type of coupling. The generality of this chimera state is confirmed with a modified integrate-and-fire model.

Self-organization phenomena that lead to pattern formation and synchronization play an important role in biology, chemistry, physics and technological applications. They manifest for example in electrical waves on the heart muscle, fireflies flashing in unison, power-grid blackouts and neurological functions as well as disorders [1]. We will present a versatile experimental setup based on optically coupled catalytic micro-particles, that allows studying synchronization patterns in very large networks of relaxation oscillators under well-controlled laboratory conditions. In particular we will show the experimental verification of the elusive spiral wave chimera state, whose existence was predicted more than 15 years ago [2]. This pattern features a wave rotating around a spatially extended core that consists of phase-randomized oscillators. [3] Taking the time delay τ in the coupling as a bifurcation parameter allows for exploring the phase diagram of spiral wave chimeras. We found that spiral wave chimeras vanish via a core expansion that is preceded by core splitting events [4]. The spiral wave chimera is likely to play a role in cortical cell ensembles, arrays of SQUIDS and carpets of cilia. [1] A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, 2001). [2] Y. Kuramoto, in Nonlinear Dynamics and Chaos: Where do we go from here? (CRC Press, 2002), p. 209–227. [3] J. F. Totz, J. Rode, M. R. Tinsley, K. Showalter, H. Engel, Spiral wave chimera states in large populations of coupled chemical oscillators. Nat. Phys. 14, 282–285 (2018). [4] J. F. Totz, M. R. Tinsley, H. Engel , K. Showalter, in preparation

We report the observation of a novel and non-trivial synchronization state in a system consisting of three oscillators coupled in a linear chain. For certain ranges of coupling strength the weakly coupled oscillator pair exhibits phase synchronization while the strongly coupled oscillator pair does not. This intriguing "weak-winner" synchronization phenomenon can be explained by the interplay between non-isochronicity and natural frequency of the oscillator, as coupling strength is varied. Further, we present sufficient conditions under which the weak-winner phase synchronization can occur for limit cycle as well as chaotic oscillators. Employing a model system from ecology as well as a paradigmatic model from physics, we demonstrate that this phenomenon is a generic feature for a large class of coupled oscillator systems. The realization of this peculiar yet quite generic weak-winner dynamics can have far reaching consequences in a wide range of scientific disciplines that deal with the phenomenon of phase synchronization including synchronization of networks. Our results also highlight the role of non-isochronicity (shear) as a fundamental feature of an oscillator in shaping the emergent dynamics.

In large networks of globally coupled BZ micro-oscillators, the synchronization transition turns out to be a discontinuous first order non-equilibrium phase transition with hysteresis. The experimental results agree well with numerical results obtained using a modified Zhabotinsky-Buchholtz-Kiyatkin-Epstein model for the chemical kinetics of the BZ reaction which accounts for photochemical coupling between the micro-oscillators. Experiments and numerical simulations indicate that the first order transition is robust with respect to changes in the frequency distribution of the uncoupled oscillators and the network connectivity. These indications suggest that the relaxation character of the individual oscillations is the essential parameter that determines the order of the synchronization transition in the network. Numerical simulations with FitzHugh-Nagumo oscillators support this hypothesis. For small values of the time-scale separation $\epsilon$ in the uncoupled oscillations, the network undergoes a reversible second order synchronization transition as weakly coupled phase oscillators in the Kuramoto model do. In contrast to that for sufficiently large $\epsilon$-values corresponding to developed relaxation oscillations the onset of synchronization takes place via a discontinuous first order transition with hysteresis. The results might be interesting for the understanding of the synchronization transition in neural networks.

In this paper, we have investigated the collective dynamical behaviors of a network of identical Hindmarsh-Rose neurons that are coupled under small-world schemes upon addition of alpha-stable Levy noise. According to the ring patterns of each neuron, we distinguish the neuronal network into spike state, burst state and spike-burst state coexistence of the neuron with both spike ring pattern and burst firing pattern. Moreover, the strength of burst is proposed to identify the firing states of the system. Furthermore, an interesting phenomenon is observed that the system presents coherence resonance in time and chimera states in space, namely coherence resonance chimeras. In addition, we show the in influences of alpha-stable Levy noise (noise intensity and stable parameter) and the small-world network (the rewiring probability) on the spike-burst state and coherence-resonance chimeras. We find that the stable parameter and noise intensity of the alpha-stable noise play a crucial role in determining the coherence-resonance chimeras and spike-burst state of the system.

We consider a one-dimensional oscillatory medium with a coupling through a diffusive linear field. In the limit of fast diffusion, this setup reduces to the classical Kuramoto-Battogtokh model. We demonstrate that for a finite diffusion stable chimera solitons, namely localized synchronous domain in an infinite asynchronous environment, are possible. The solitons are stable also for finite density of oscillators, but in this case they sway with a nearly constant speed. This finite-density-induced motility disappears in the continuum limit, as the velocity of the solitons is inverse proportional to the density. A long-wave instability of the homogeneous asynchronous state causes soliton turbulence, which appears as a sequence of soliton mergings and creations. As the instability of the asynchronous state becomes stronger, this turbulence develops into a spatio-temporal intermittency.

Networks of coupled oscillators exhibit a variety of dynamical synchronization behaviors that are of significance in nature and technology. The emergence of chimera states - patterns with coexisting coherent and incoherent domains - has been demonstrated recently in experiments using mechanical oscillators [Martens et al., Proc. Natl. Acad. Sci., 110 (2013)]. Here, I provide an intuitive explanation for the emergence of chimera states based on fundamental mechanical principles, identifying resonance as a physical mechanism leading to chimera states. Thereby, I establish a rigorous link between the mechanical model presented in Martens et al. (2013) and abstract mathematical models in which chimera states were discovered originally, i.e., (i) Stuart-Landau oscillators [Laing, Phys. Rev. E, 81 (2010).] and (ii) Kuramoto oscillators [Abrams et al., Phys. Rev. Lett., 101 (2008); Martens et al., Chaos, 26 (2016); Bick et al., Chaos, 28 (2018)], and we explore for what parameter regimes such a mapping is valid. The resulting mechanical theory likely translates to a large range of domains in nature and technology.

Chimera states are an intriguing example of partial synchronization patterns emerging in networks of identical oscillators. They consist of spatially coexisting domains of coherent (synchronized) and incoherent (desynchronized) dynamics. We show that a plethora of chimera patterns arise if one goes beyond the Kuramoto phase oscillator model, and considers coupled phase and amplitude dynamics, and more complex topologies than a simple one-dimensional ring network, e.g., fractal connectivities or multi-layer structures. For the generic FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator various multi-chimera patterns including amplitude-mediated phase chimeras, amplitude chimeras, chimera death, and double chimeras occur. We review the control of chimera patterns by a subtle interplay of dynamics, topology, and delay. Our focus is in particular on partial relay synchronization in brain networks, and on applications to brain dynamics with empirical connectivities like epileptic seizure and unihemispheric sleep.

In this talk, I will report on various chimera-like states in dynamic adaptive networks, including the adaptive Kuramoto model as well as more complex neuronal networks. The observed states are characterized by the different average frequencies of oscillations in the network. The distribution of average frequencies can be delta-shaped, leading to clusters or "weak chimeras", or continuous, resembling classical chimera states.

I present here formation of chimera states in three different arrays of the superconducting Josephson junctions. In an array of identical Josephson junctions under global mean field repulsive interactions, formation of chimera states has been observed. In a different network of heterogeneous Josephson junctions under repulsive global coupling, we observe a chimera-like pattern with coexisting groups of coherent junctions in librational motion, another group of incoherent junctions in rotational motions and a third group of incoherent junctions that show intermittent large events which we identify as creating extreme events. Finally, we find a smallest set of three identical junctions which achieves complete synchronization under repulsive coupling. However, for stronger repulsive interactions, they break into two groups, two oscillators maintain coherence while third one goes to incoherence. All the results are based on numerical experiments.

Superconducting Quantum Interference Devices, or SQUIDS, are building blocks of quantum devices. They can form metamaterials with unique dynamical and spectral properties [1]. Chimera states may be generated in SQUID metamaterials leading to spatiotemporal organisation, new resonant properties and enhanced functionality [2, 3]. We present theoretical and numerical work that points to the possible role of chimeras in SQUID-based extended systems [4]. Additionally, we show that the application of machine learning techniques such as reservoir computing and Long-Short Term Memory (LSTM) networks may enhance the utility and control of chimera states in superconducting metamaterials. References [1] rf SQUID metamaterials, N. Lazarides and G. P. Tsironis, Appl. Phys. Lett.90, 163501 (2007) [2] N. Lazarides, G. Neofotistos, and G. P. Tsironis, “ Chimeras in SQUID Metamaterials ”, Physical Review B 91, 054303 (2015) [3] J. Hizanidis, N. Lazarides, and G. Tsironis: “Robust chimera states in SQUID metamaterials with local interactions”, Phys. Rev. E 94, 032219 (2016) [4] N. Lazarides nd G. P. Tsironis, Superconducting metamaterials, Phys. Rep. 752, 1 (2018) [5] G. Neofotistos, M. Mattheakis, G. D. Barmparis, J. Hizanidis, G. P. Tsironis and E. Kaxiras, Machine learning with observers predicts complex spatiotemporal behavior, Front. Phys. 7 ,24, (2019)

We show that amplitude-mediated phase chimeras and amplitude chimeras can occur in the same network of nonlocally coupled identical oscillators. These are two different partial synchronization patterns, where spatially coherent domains coexist with incoherent domains and coherence/incoherence referring to both amplitude and phase or only the amplitude of the oscillators, respectively. By changing the coupling strength, the two types of chimera patterns can be induced. We find numerically that the amplitude chimeras are not short-living transients but can have a long lifetime. Also, we observe variants of the amplitude chimeras with quasiperiodic temporal oscillations. We provide a qualitative explanation of the observed phenomena in the light of symmetry breaking bifurcation scenarios. We believe that this study will shed light on the connection between two disparate chimera states having different symmetry-breaking properties.

Despite their simplicity, networks of coupled phase oscillators can give rise to intriguing collective dynamical phenomena. However, the symmetries of globally and identically coupled identical units do not allow solutions where distinct oscillators are frequency-unlocked—a necessary condition for the emergence of chimeras. Thus, forced symmetry breaking is necessary to observe chimera-type solutions. Here, we consider the bifurcations that arise when full permutational symmetry is broken for the network to consist of coupled populations. We consider the smallest possible network composed of four phase oscillators and elucidate the phase space structure, (partial) integrability for some parameter values, and how the bifurcations away from full symmetry lead to frequency-unlocked weak chimera solutions. Since such solutions wind around a torus they must arise in a global bifurcation scenario. Moreover, periodic weak chimeras undergo a period doubling cascade leading to chaos. The resulting chaotic dynamics with distinct frequencies do not rely on amplitude variation and arise in the smallest networks that support chaos

Stability properties of identical nonlocally coupled phase oscillators on periodic lattices in two and three dimensions are studied using a combination of analytical and numerical techniques. The problem is formulated in terms of the Ott-Antonsen description valid in the continuum limit and applied to a number of different chimera states including spiral chimeras in two dimensions and coherent ball chimeras in three dimensions. The analytical predictions are confirmed using direct numerical simulations. References: O.E. Omel'chenko, M. Wolfrum and E. Knobloch, SIADS 17, 87-127 (2018) O.E. Omel'chenko and E. Knobloch, New J. Phys. 21, 093034 (2019)

Complex-valued quadratic maps either converge to fixed points, enter into periodic cycles, show aperiodic behavior, or diverge to infinity. Which of these scenarios takes place depends on the map’s complex-valued parameter c and the initial conditions. The Mandelbrot set is defined by the set of c values for which the map remains bounded when initiated at the origin of the complex plane. In this study, we analyze the dynamics of a coupled network of two pairs of two quadratic maps in dependence on the parameter c. Across the four maps, c is kept the same whereby the maps are identical. In analogy to the behavior of individual maps, the network iterates either diverge to infinity or remain bounded. The bounded solutions settle into different stable states, including full synchronization and desynchronization of all maps. Furthermore, symmetric partially synchronized states of within-pair synchronization and across-pair synchronization as well as a symmetry broken chimera state are found. The boundaries between bounded and divergent solutions in the domain of c are fractals showing a rich variety of intriguingly esthetic patterns. Moreover, the set of bounded solutions is divided into countless subsets throughout all length scales in the complex plane. Each individual subset contains only one state of synchronization and is enclosed within fractal boundaries by c values leading to divergence.

Sleep dynamics have been described as a switching between two broad states; non-rapid eye movement sleep (nREM), which is thought to originate from widespread neuronal synchronization, and REM, with neuronal desynchronization reminiscent of waking behaviour. While it was thought that REM and nREM constituted global brain states, recent research suggests that this is not true --- spatio-temporally localized neuronal synchronization and desynchronization coexist, produced by complex neuronal dynamics during sleep. Here, we utilize time-frequency analysis of mesoscopic voltage-sensitive dye recordings of anesthetized mice to study the evolution of neuronal desynchronization across the cortex in space and time. We find variations in the spatial distribution of desynchronization and propagation which mimics a critical spreading process. This suggests an operating regime at an 'edge-of-synchronization' phase transition, and dynamics reminiscent of coexisting phase synchronization and desynchronization in chimera states.

Chimeras occur in networks of two coupled populations of oscillators when the oscillators in one population synchronize while those in the other are asynchronous. We consider chimeras of this form in networks of planar oscillators for which one parameter associated with the dynamics of an oscillator is randomly chosen from a uniform distribution. A generalization of the previous approach [Laing, Phys. Rev. E 100, 042211 (2019)], which dealt with identical oscillators, is used to investigate the existence and stability of chimeras for these heterogeneous networks in the limit of an infinite number of oscillators. In all cases, making the oscillators more heterogeneous destroys the stable chimera in a saddle-node bifurcation. The results help us understand the robustness of chimeras in networks of general oscillators to heterogeneity.

Chimera states have been a vibrant subject of research in the recent past, but the analytical treatment of transitions from chimeras to coherent states remains a challenge. Here we analytically derive the necessary conditions for this transition by means of the coherent stability function approach, which is akin to the master stability function approach that is traditionally used to study the stability of synchronization in coupled oscillators. In chimera states, there is typically at least one group of oscillators that evolves in a drifting, random manner, while other groups of oscillators follow a smoother, more coherent profile. In the coherent state, there thus exists a smooth functional relationship between the oscillators. This lays the foundation for the coherent stability function approach, where we determine the stability of the coherent state. We subsequently test the analytical prediction numerically by calculating the strength of incoherence during the transition point. We use leech neurons, which exhibit coexistence of chaotic and periodic tonic spiking depending on initial conditions, coupled via nonlocal electrical synapses, to demonstrate our approach. We systematically explore various dynamical states with the focus on the transitions between chimeras and coherence, fully confirming the validity of the coherent stability function. We also observe complete synchronization for higher values of the coupling strength, which we verify by the master stability function approach. Recently, we investigate that higher-order interaction promotes chimera states. Ref. 1. Sarbendu Rakshit, Zahra Faghani, Fatemeh Parastesh, Shirin Panahi, Sajad Jafari, Dibakar Ghosh, and Matjaž Perc, Physical Review E 100, 012315 (2019). 2. Srilena Kundu and Dibakar Ghosh, Higher-order interaction promotes chimera states (to be published) 2022.

I will discuss a unifying perspective to the multitude of actors that operate in today's and tomorrow's power grids. These exhibit a wide range of exotic states that need to be prevented and controlled against in order to achieve a stable self-organized equilibrium. I show how a number of probabilistic concepts and methods can help understand the proliferation of states that occur.

Chimera states occur widely in networks of identical oscillators as it has been shown in the recent extensive theoretical and experimental research. In such a state, different groups of oscillators can exhibit co–existing synchronous and incoherent behaviors despite homogeneous coupling. Here we consider a star network, in which N identical peripheral end nodes are connected to the central hub node. We find that if a single node exhibits transient chaotic behavior in the whole network, the pattern of transient chimera–like state, which persists for a significant amount of time, is created. As a proof of the concept, we examine the system of N double pendula (peripheral end nodes) located on the periodically oscillating platform (central hub). We show that such, transient chimera–like states can be observed in simple experiments with mechanical oscillators, which are controlled by elementary dynamical equations. Our finding suggests that transient chimera–like states are observable in networks relevant to various real–world systems.