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 {\it 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 hundred oscillators (cyclic century), the islands merge into a single {\it chimera continent} which incorporates the world of chimeras of different configuration. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as it decreases.
University of Namur（南纳穆尔大学）；Institute of Science Tokyo (former Tokyo Tech)（东京科学大学（原东京工业大学））；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心 (iTHEMS)）
We suggest a definition for a type of chimera state that appears in networks of indistinguishable phase oscillators. Defining a "weak chimera" as a type of invariant set showing partial frequency synchronization, we show that this means they cannot appear in phase oscillator networks that are either globally coupled or too small. We exhibit various networks of four, six and ten indistinguishable oscillators where weak chimeras exist with various dynamics and stabilities. We examine the role of Kuramoto-Sakaguchi coupling in giving degenerate (neutrally stable) families of weak chimera states i…
Chimera states in spatially extended networks of oscillators have some oscillators synchronised while the remainder are asynchronous. These states have primarily been studied in networks with nonlocal coupling, and more recently in networks with global coupling. Here we present three networks with only local coupling (diffusive, to nearest neighbours) which are numerically found to support chimera states. One of the networks is analysed using a self-consistency argument in the continuum limit, and this is used to find the boundaries of existence of a chimera state in parameter space.
A chimera state is a spatiotemporal pattern in which a network of identical coupled oscillators exhibits coexisting regions of asynchronous and synchronous oscillation. Two distinct classes of chimera states have been shown to exist: "spots" and "spirals." Here we study coupled oscillators on the surface of a sphere, a single system in which both spot and spiral chimera states appear. We present an analysis of the birth and death of spiral chimera states and show that although they coexist with spot chimeras, they are stable in disjoint regions of parameter space.
We found that a network-organized metapopulation of cooperators, defectors and destructive agents playing the public goods game with mutations, can collectively reach global synchronization or chimera states. Global synchronization is accompanied by a collective periodic burst of cooperation, whereas chimera states reflect the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation. Numerical simulations have shown that the system's dynamics alternates between these two steady states through a first order transition. Depending o…
We report the diversity of scroll wave chimeras in the three-dimensional (3D) Kuramoto model with inertia for N^{3} 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 this 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 ch…
Recently a novel dynamical state, called the {\it chimera death}, is discovered in a network of non locally coupled identical oscillators [A. Zakharova, M. Kapeller, and E. Schöll, Phy.Rev.Lett. 112, 154101 (2014)], which is defined as the coexistence of spatially coherent and incoherent oscillation death state. This state arises due to the interplay of non locality and symmetry breaking and thus bridges the gap between two important dynamical states, namely the chimera and oscillation death. In this paper we show that the chimera death can be induced in a network of generic identical oscillat…
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 state…
We find chimera states with respect to amplitude dynamics in a network of Stuart-Landau oscillators. These partially coherent and partially incoherent spatio-temporal patterns appear due to the interplay of nonlocal network topology and symmetry-breaking coupling. As the coupling range is increased, the oscillations are quenched, amplitude chimeras disappear and the network enters a symmetry-breaking stationary state. This particular regime is a novel pattern which we call chimera death. It is characterized by the coexistence of spatially coherent and incoherent inhomogeneous steady states and…
We consider a social-type network of coupled phase oscillators. Such a network consists of an active core of mutually interacting elements, and of a flock of passive units, which follow the driving from the active elements, but otherwise are not interacting. We consider a ring geometry with a long-range coupling, where active oscillators form a fluctuating chimera pattern. We show that the passive elements are strongly correlated. This is explained by negative transversal Lyapunov exponents.
We consider star networks of chaotic oscillators, with all end-nodes connected only to the central hub node, under diffusive coupling, conjugate coupling and mean-field type coupling. We observe the existence of chimeras in the end-nodes, which are identical in terms of the coupling environment and dynamical equations. Namely, the symmetry of the end-nodes is broken and co-existing groups with different synchronization features and attractor geometries emerge. Surprisingly, such chimera states are very wide-spread in this network topology, and large parameter regimes of moderate coupling stren…
A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.
We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generally difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized.
We investigate the emergence of amplitude and frequency chimera states in ring-star networks consisting of identical Chua circuits connected via nonlocal diffusive, bidirectional coupling. We first identify single-well chimera patterns in a ring network under nonlocal coupling schemes. When a central node is added to the network, forming a ring-star network, the central node acts as the distributor of information, increasing the chances of synchronization. Numerical simulations show that the radial coupling strength $k$ between the central and the peripheral nodes acts as an order parameter le…
For a globally coupled network of semiconductor lasers with delayed optical feedback, we demonstrate the existence of chimera states. The domains of coherence and incoherence that are typical for chimera states are found to exist for the amplitude, phase, and inversion of the coupled lasers. These chimera states defy several of the previously established existence criteria. While chimera states in phase oscillators generally demand nonlocal coupling, large system sizes, and specially prepared initial conditions, we find chimera states that are stable for global coupling in a network of only fo…
We show that chimera states, where differentiated subsets of synchronized and desynchronized dynamical elements coexist, can emerge in networks of hyperbolic chaotic oscillators subject to global interactions. As local dynamics we employ Lozi maps which possess hyperbolic chaotic attractors. We consider a globally coupled system of these maps and use two statistical quantities to describe its collective behavior: the average fraction of elements belonging to clusters and the average standard deviation of state variables. Chimera states, clusters, complete synchronization, and incoherence are t…
Chimera is a relatively new emerging phenomenon where coexistence of synchronous and asynchronous state is observed in symmetrically coupled dynamical units. We report observation of the chimera state in multiplex networks where individual layer is represented by 1-d lattice with non-local interactions. While, multiplexing does not change the type of the chimera state and retains the multi-chimera state displayed by the isolated networks, it changes the regions of the incoherence. We investigate emergence of coherent-incoherent bifurcation upon varying the control parameters, namely, the coupl…
In this work we present a method to control chimera states through linear augmentation. Using an ensemble of globally coupled oscillators, we demonstrate that control over the spatial location of the incoherent or coherent regions of a chimera state in a network can be achieved. This is characterized by exploring their basins of attraction. We also verify the stability of different states present in the coupled systems before and after linear augmentation by calculating the transverse Lyapunov exponent and Master stability function. We observe that the control through this technique is indepen…
Nontrivial collective behavior may emerge from the interactive dynamics of many oscillatory units. Chimera states are chaotic patterns of spatially localized coherent and incoherent oscillations. The recently-introduced notion of a weak chimera gives a rigorously testable characterization of chimera states for finite-dimensional phase oscillator networks. In this paper we give some persistence results for dynamically invariant sets under perturbations and apply them to coupled populations of phase oscillators with generalized coupling. In contrast to the weak chimeras with nonpositive maximal …
We present results obtained for a network of four delay-coupled lasers modelled by Lang-Kobayashi-type equations. We find small chimera states consisting of a pair of synchronized lasers and two unsynchronized lasers. One class of these small chimera states can be understood as intermediate steps on the route from synchronization to desynchronization and we present the entire chain of bifurcations giving birth to them. This class of small chimeras can exhibit limit-cycle or quasiperiodic dynamics. A second type of small chimera states exists apparently disconnected from any region of synchroni…
The notion of a weak chimeras provides a tractable definition for chimera states in networks of finitely many phase oscillators. Here we generalize the definition of a weak chimera to a more general class of equivariant dynamical systems by characterizing solutions in terms of the isotropy of their angular frequency vector - for coupled phase oscillators the angular frequency vector is given by the average of the vector field along a trajectory. Symmetries of solutions automatically imply angular frequency synchronization. We show that the presence of such symmetries is not necessary by giving…
Symmetry breaking spatial patterns, referred to as chimera states, have recently been catapulted into the limelight due to their coexisting coherent and incoherent hybrid dynamics. Here, we present a method to engineer a chimera state by using an appropriate distribution of heterogeneous time delays on the edges of a network. The time delays in interactions, intrinsic to natural or artificial complex systems, are known to induce various modifications in spatiotemporal behaviors of the coupled dynamics on networks. Using a coupled chaotic map with the identical coupling environment, we demonstr…
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 refer 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 varia…
Chimera referring to a coexistence of coherent and incoherent states, is traditionally very difficult to control due to its peculiar nature. Here, we provide a recipe to construct chimera states in the multiplex networks with the aid of multiplexing-delays. The chimera state in multiplex networks is produced by introducing heterogeneous delays in a fraction of inter-layer links, referred as multiplexing-delay, in a sequence. Additionally, the emergence of the incoherence in the chimera state can be regulated by making appropriate choice of both inter- and intra-layer coupling strengths, wherea…
We study the existence of chimera states in pulse-coupled networks of bursting Hindmarsh-Rose neurons with nonlocal, global and local (nearest neighbor) couplings. Through a linear stability analysis, we discuss the behavior of stability function in the incoherent (i.e. disorder), coherent, chimera and multi-chimera states. Surprisingly, we find that chimera and multi-chimera states occur even using local nearest neighbor interaction in a network of identical bursting neurons alone. This is in contrast with the existence of chimera states in populations of nonlocally or globally coupled oscill…
In this paper we report the occurrence of chimera patterns in a network of neuronal oscillators, which are coupled through {\it local}, synaptic {\it gradient} coupling. We discover a new chimera pattern, namely the {\it imperfect traveling chimera} where the incoherent traveling domain spreads into the coherent domain of the network. Remarkably, we also find that chimera states arise even for {\it one-way} local coupling, which is in contrast to the earlier belief that only nonlocal, global or nearest neighbor local coupling can give rise to chimera; this find further relaxes the essential co…
In a network of pulse-coupled oscillators with adaptive coupling, we a dynamical regime which we call an `itinerant chimera'. Similarly as in classical chimera states, the network splits into two domains, the coherent and the incoherent ones. The drastic difference is that the composition of the domains is volatile, i.e. the oscillators demonstrate spontaneous switching between the domains. This process can be seen as traveling of the oscillators from one domain to another, or as traveling of the chimera core across network. We explore the basic features of the itinerant chimeras, such as the …
In a recent study of chaos synchronization in symmetric complex networks [Pecora \textit{et al}., Nat. Commun. {\bf 5}, 4079 (2014)], it is found that stable synchronous clusters may coexist with many non-synchronous nodes in the asynchronous regime, resembling the chimera state observed in regular networks of non-locally coupled periodic oscillators. Although of practical significance, this new type of state, namely the chimera-synchronization state, is hardly generated for the general complex networks, due to either the topological instabilities or the weak coupling strength. Here, based on …
We study the stable spatiotemporal patterns that arise in a 3D network of neuron oscillators, whose dynamics is described by the Leaky Integrate-and-Fire (LIF) model. More specifically, we investigate the form of the chimera states induced by a 3D coupling matrix with nonlocal topology. The observed patterns are in many cases direct generalizations of the corresponding 2D patterns, e.~g. spheres, layers and cylinder grids. We also find cylindrical and "cross-layered" chimeras that do not have an equivalent in 2D systems. Quantitative measures are calculated, such as the ratio of synchronized a…
We investigate spatio-temporal patterns occurring in a two-layer multiplex network of oscillatory FitzHugh-Nagumo neurons, where each layer is represented by a nonlocally coupled ring. We show that weak multiplexing, i.e., when the coupling between the layers is smaller than that within the layers, can have a significant impact on the dynamics of the neural network. We develop control strategies based on weak multiplexing and demonstrate how the desired state in one layer can be achieved without manipulating its parameters, but only by adjusting the other layer. We find that for coupling range…
Chimera states have been recently found in a variety of different coupling schemes and geometries. In most cases, the underlying coupling structure is considered to be static, while many realistic systems display significant temporal changes in the pattern of connectivity. In this work, we investigate a time-varying network made of two coupled populations of Kuramoto oscillators, where the links between the two groups are considered to vary over time. As a main result, we find that the network may support stable, breathing and alternating chimera states. We also find that, when the rate of con…
We observe chimeralike states in an ensemble of oscillators using a type of global coupling consisting of two components: attractive and repulsive mean-field feedback. We identify existence of two types of chimeralike states in a bistable Liénard system; in one type, both the coherent and the incoherent populations are in chaotic states (called as chaos-chaos chimeralike states) and, in another type, the incoherent population is in periodic state while the coherent population has irregular small oscillation. Interestingly, we also recorded a metastable state in a parameter regime of the Liénar…
Chimera states, which consist of coexisting synchronous and asynchronous domains in networks of coupled oscillators, are in the focus of attention for over a decade. Although chimera morphology and properties have been investigated in a number of models, the mechanism responsible for their formation is still not well understood. To shed light in the chimera producing mechanism, in the present study we introduce an oscillatory model with variable frequency governed by a 3rd order equation. In this model single oscillators are constructed as bistable and depending on the initial conditions their…
We report a novel spatiotemporal state, namely the chimera-like incongruous coexistence of {\it synchronized oscillation} and {\it stable steady state} (CSOD) in a realistic ecological network of nonlocally coupled oscillators. Unlike the {\it chimera} and {\it chimera death} state, in the CSOD state identical oscillators are self-organized into two coexisting spatially separated domains: In one domain neighboring oscillators show synchronized oscillation and in another domain the neighboring oscillators randomly populate either a synchronized oscillating state or a stable steady state (we cal…
Symmetry broken states arise naturally in oscillatory networks. In this Letter, we investigate chaotic attractors in an ensemble of four mean-coupled Stuart-Landau oscillators with two oscillators being synchronized. We report that these states with partially broken symmetry, so-called chimera states, have different set-wise symmetries in the incoherent oscillators, and in particular some are and some are not invariant under a permutation symmetry on average. This allows for a classification of different chimera states in small networks. We conclude our report with a discussion of related stat…
Chimera states -- named after the mythical beast with a lion's head, a goat's body, and a dragon's tail -- correspond to spatiotemporal patterns characterised by the coexistence of coherent and incoherent domains in coupled systems. They were first identified in 2002 in theoretical studies of spatially extended networks of Stuart-Landau oscillators, and have been subject to extensive theoretical and experimental research ever since. While initially considered peculiar to networks with weak nonlocal coupling, recent theoretical studies have predicted that chimera-like states can emerge even in …
Classical chimera states are paradigmatic examples of partial synchronization patterns emerging in nonlinear dynamics. These states are characterized by the spatial coexistence of two dramatically different dynamical behaviors, i.e., synchronized and desynchronized dynamics. Our aim in this contribution is to discuss signatures of chimera states in quantum mechanics. We study a network with a ring topology consisting of N coupled quantum Van der Pol oscillators. We describe the emergence of chimera-like quantum correlations in the covariance matrix. Further, we establish the connection of chim…
Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robus…
We study the spatiotemporal dynamics of a ring of nonlocally coupled FitzHugh-Nagumo oscillators in the bistable regime. A new type of chimera patterns has been found in the noise-free network and when isolated elements do not oscillate. The region of existence of these structures has been explored when the coupling range and the coupling strength between the network elements are varied.
In this article, we report the development of an emerging dynamical state, namely, the alternating chimera, in a network of identical neuronal systems induced by an external electromagnetic field. Owing to this interaction scenario, the nonlinear neuronal oscillators are coupled indirectly via electromagnetic induction with magnetic flux, through which neurons communicate in spite of the absent physical connections among them. The evolution of each neuron, here, is described by the three-dimensional Hindmarsh-Rose dynamics. We demonstrate that the presence of such non-locally and globally inte…
In this paper, we have introduced and investigated the collective behavior of a network of memristive Hindmarsh-Rose (HR) neurons. The proposed model was built considering the memristive autapse of the traditional 2D HR neuron. Using the one-parameter bifurcation diagram and its corresponding maximal Lyapunov exponent graph, we showed that the proposed model was able to exhibit a reverse period doubling route to chaos, phenomenon of interior and exterior crises. Three different configurations of the ring-star network of the memristive HR neuron model, including ring-star, ring, and star, have …
Indian Institute of Science Education and Research (IISER), Pune（印度科学教育研究所（浦那））；Indian Institute of Science Education and Research (IISER), Thiruvananthapuram（印度科学教育研究所（特里凡得琅））；National Brain Research Centre（国家脑研究中心）；Heyrovsky Institute, Czech Academy of Sciences（捷克科学院海罗夫斯基研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）
We present the emergence of chimeras, a state referring to coexistence of partly coherent, partly incoherent dynamics in networks of identical oscillators, in a multiplex network consisting of two non-identical layers which are interconnected. We demonstrate that the parameter range displaying the chimera state in the homogeneous first layer of the multiplex networks can be tuned by changing the link density or connection architecture of the same nodes in the second layer. We focus on the impact of the interconnected second layer on the enlargement or shrinking of the coupling regime for which…
Chimera states, namely the coexistence of coherent and incoherent behavior, were previously analyzed in complex networks. However, they have not been extensively studied in modular networks. Here, we consider the neural network of the \textit{C.elegans} soil worm, organized into six interconnected communities, where neurons obey chaotic bursting dynamics. Neurons are assumed to be connected with electrical synapses within their communities and with chemical synapses across them. As our numerical simulations reveal, the coaction of these two types of coupling can shape the dynamics in such a wa…
Digital phase-locked loops (DPLLs) are nonlinear feedback-controlled systems that are widely used in electronic communication and signal processing applications. In most of the applications they work in coupled mode, however, vast of the studies on DPLLs concentrate on the dynamics of a single isolated unit. In this paper we consider both one- and two-dimensional networks of DPLLs connected through a practically realistic nonlocal coupling and explore their collective dynamics. For the one-dimensional network we analytically derive the parametric zone of stable phase-locked state in which DPLL…
Coherence-resonance chimera was discovered in [Phys. Rev. Lett. 117, 014102 (2016)], which combines the effect of coherence resonance and classical chimeras in the presence of noise in a network of type-II excitable systems. However, the same in a network of type-I excitable units has not been observed yet. In this paper, for the first time, we report the occurrence of coherence-resonance chimera in coupled type-I excitable systems. We consider a paradigmatic model of type-I excitability, namely the saddle-node infinite period model and show that the coherence-resonance chimera appears over an…
