Types Of Chimera States Research Articles

Recovery of synchronized oscillations on multiplex networks by tuning dynamical time scales

The heterogeneity among interacting dynamical systems or variations in the pattern of their interactions occur naturally in many real complex systems. Often they lead to partially synchronized states like chimeras or oscillation suppressed states like in-homogeneous or homogeneous steady states. In such cases, it is a challenge to get synchronized oscillations in spite of prevailing heterogeneity. In this study, we present a formalism for controlling multi layer, multi timescale systems and show how synchronized oscillations can be restored by tuning the dynamical time scales between the layers. Specifically, we use the model of a multiplex network, where the first layer of coupled oscillators is multiplexed with an environment layer, that can generate various types of chimera states and suppressed states. We show that by tuning the time scale mismatch between the layers, we can revive the synchronized oscillations. We analyse the nature of the transition of the system to synchronization from various dynamical states and the role of time scale mismatch and strength of inter layer coupling in this scenario. We also consider a three-layer multiplex system, where two system layers interact with the common environment layer. In this case, we observe anti synchronization and in-homogeneous steady states on the system layers and by tuning their time scale difference with the environment layer, they undergo transition to synchronized oscillations.

Synchronization effects in multiplex networks of chaotic maps with memristive interlayer coupling

Models of two-layer multiplex networks of one-dimensional maps with memristive interlayer coupling are studied. In the absence of coupling, the layers exhibit different types of chimera states. Two types of maps are considered as the elements of networks: modified Ricker maps and cubic maps. The effects of complete and partial synchronization of the spatiotemporal dynamics of layers are observed in the case of a homogeneous and inhomogeneous network, respectively. The dependence of the synchronization threshold for the coupling parameter on the initial states of the memristors has been established. This dependence is partially preserved in the case of non-ideal memristors. Diagrams of the synchronization error are constructed on the plane of the initial state variable of memristors and the interlayer coupling coefficient. The effect of noise in memristive coupling elements on synchronization is considered.

Noise-modulated nonlocal coupling: Chimera suppression and beyond

The spatio-temporal dynamics of networks of chaotic discrete-time systems is studied numerically in the presence of colored Gaussian noise-modulated nonlocal coupling between elements. As individual nodes of the networks, a modified Ricker map and a cubic map are used. Without noise, the networks demonstrate various types of chimera states, such as phase or double-well chimeras. The nonlocal coupling is modulated by either a common noise source or independent noise sources. The impact of noise-modulated coupling (coupling noise) on the dynamics of the networks is analyzed for a number of different random initial conditions and noise realizations. It is shown that the network dynamics and settled spatio-temporal structures can be controlled by varying the intensity and spectral characteristics of the coupling noise.

Multistability and evolution of chimera states in a network of type II Morris-Lecar neurons with asymmetrical nonlocal inhibitory connections.

Formation of synchronous activity patterns is an essential property of neuronal networks that has been of central interest to synchronization theory. Chimera states, where both synchronous and asynchronous activities of neurons co-exist in a single network, are particularly poignant examples of such patterns, whose dynamics and multistability may underlie brain function, such as cognitive tasks. However, dynamical mechanisms of coherent state formation in spiking neuronal networks as well as ways to control these states remain unclear. In this paper, we take a step in this direction by considering the evolution of chimera states in a network of class II excitable Morris-Lecar neurons with asymmetrical nonlocal inhibitory connections. Using the adaptive coherence measure, we are able to partition the network parameter space into regions of various collective behaviors (antiphase synchronous clusters, traveling waves, different types of chimera states as well as a spiking death regime) and have shown multistability between the various regimes. We track the evolution of the chimera states as a function of changed key network parameters and found transitions between various types of chimera states. We further find that the network can demonstrate long transients leading to quasi-persistence of activity patterns in the border regions hinting at near-criticality behaviors.

Amplitude-mediated chimera states in nonlocally coupled Stuart-Landau oscillators.

Chimera states achieve the coexistence of coherent and incoherent subgroups through symmetry breaking and emerge in physical, chemical, and biological systems. We show the presence of amplitude-mediated multicluster chimera states in nonlocally coupled Stuart-Landau oscillators. We clarify the prerequisites for having different types of chimera states by analytically and numerically studying how phase transitions occur between these states. Our results demonstrate how the oscillation amplitudes interact with the phase degrees of freedom in chimera states and significantly advance our understanding of the generation mechanisms of such states in coupled oscillator systems.

Transition of Chimera States and Synchronization in Two-Layer Networks of Coupled Hindmarsh–Rose Neurons

The present work focuses on the existence of chimera states in two-layer networks of locally, nonlocally, and globally coupled Hindmarsh–Rose neurons. We show that chimera states occur in all three coupled neuronal networks by changing synaptic coupling strength and the number of coupled neighbors, and traveling chimera state exists in networks by using local and nonlocal couplings. Particularly, we find that in the nonlocally coupled network, there exist regions of coherent state, which can produce a wave such that the period becomes longer with the increase of the number of coupled neighbors. Interestingly, we observe a new chimera state with the coexistence of interlayer synchronous and asynchronous chimera states, which we named as the interlayer semi-synchronous chimera state. In addition, the results indicate that the interaction between layers in neuronal networks can induce different types of chimera states and firing patterns, which could be useful to control neural firing patterns and deepen the understanding of neuronal evolution where coherent and incoherent dynamics coexist.

Attracting Poisson chimeras in two-population networks.

Chimera states, i.e., dynamical states composed of coexisting synchronous and asynchronous oscillations, have been reported to exist in diverse topologies of oscillators in simulations and experiments. Two-population networks with distinct intra- and inter-population coupling have served as simple model systems for chimera states since they possess an invariant synchronized manifold in contrast to networks on a spatial structure. Here, we study dynamical and spectral properties of finite-sized chimeras on two-population networks. First, we elucidate how the Kuramoto order parameter of the finite-sized globally coupled two-population network of phase oscillators is connected to that of the continuum limit. These findings suggest that it is suitable to classify the chimera states according to their order parameter dynamics, and therefore, we define Poisson and non-Poisson chimera states. We then perform a Lyapunov analysis of these two types of chimera states, which yields insight into the full stability properties of the chimera trajectories as well as of collective modes. In particular, our analysis also confirms that Poisson chimeras are neutrally stable. We then introduce two types of "perturbation" that act as small heterogeneities and render Poisson chimeras attracting: A topological variation via the simplest nonlocal intra-population coupling that keeps the network symmetries and the allowance of amplitude variations in the globally coupled two-population network; i.e., we replace the phase oscillators by Stuart-Landau oscillators. The Lyapunov spectral properties of chimera states in the two modified networks are investigated, exploiting an approach based on network symmetry-induced cluster pattern dynamics of the finite-size network.

Dynamics in two interacting subpopulations of nonidentical phase oscillators.

Chimera states refer to the dynamical states in which the inherent symmetry of the system is broken. The system composed of two interacting identical subpopulations of phase oscillators provides a platform to study chimera states. In this system, different types of chimera states have been identified and the transitions between them have been investigated. However, the parameter space is not fully explored in this system. In this work, we study a system comprised of two interacting subpopulations of nonidentical phase oscillators. Through numerical simulations and theoretical analyses, we find three symmetry-reserving states, including incoherent state, in-phase synchronous state, and antiphase synchronous state, and three types of symmetry-breaking states, including in-phase chimera states, antiphase chimera states, and weak chimera states. The stability diagrams of these dynamical states are explored on different parameter planes and transition scenarios amongst these states are investigated. We find that the weak chimera states act as the bridge between in-phase and antiphase chimera states. We also observe the existence of a period-two chimera state, chaotic chimera state, and drifting chimera states.

Multi-headed loop chimera states in coupled oscillators.

In this paper, we introduce a novel type of chimera state, characterized by the geometrical distortion of the coherent ring topology of coupled oscillators. The multi-headed loop chimeras are examined for a simple network of locally coupled pendulum clocks, suspended on the vertical platform. We determine the regions of the occurrence of the observed patterns, their structure, and possible co-existence. The representative examples of behaviors are shown, exhibiting the variety of configurations that can be observed. The statistical analysis of the solutions indicates the geometrical regions of the system with the highest probability of the chimeras' occurrence. We investigate the mechanism of the creation of the observed states, showing that the manipulation of the initial positions of chosen pendula may induce the desired patterns. Apart from the study of the isolated network, we also discuss the scenario of the movable platform, showing a possible influence of the global coupling structure on the stability of the observed states. The stability of loop chimeras is examined for varying both the amplitude and the frequency of the oscillations of the platform. We indicate the excitation parameters for which the solutions can survive as well as be destroyed. The bifurcation analysis included in the paper allows us to discuss the transitions between possible behaviors. The appearance of multi-headed loop chimeras is generalized into large networks of oscillators, showing the universal character of the observed patterns. One should expect to observe similar results also in other types of coupled oscillators, especially the mechanical ones.

Critical Switching in Globally Attractive Chimeras

We report on a new type of chimera state that attracts almost all initial conditions and exhibits power-law switching behavior in networks of coupled oscillators. Such switching chimeras consist of two symmetric configurations, which we refer to as subchimeras, in which one cluster is synchronized and the other is incoherent. Despite each subchimera being linearly stable, switching chimeras are extremely sensitive to noise: arbitrarily small noise triggers and sustains persistent switching between the two symmetric subchimeras. The average switching frequency scales as a power law with the noise intensity, which is in contrast with the exponential scaling observed in typical stochastic transitions. Rigorous numerical analysis reveals that the power-law switching behavior originates from intermingled basins of attraction associated with the two subchimeras, which in turn are induced by chaos and symmetry in the system. The theoretical results are supported by experiments on coupled optoelectronic oscillators, which demonstrate the generality and robustness of switching chimeras.

Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators

Abstract We study numerically autowave structures in a two-dimensional lattice of van der Pol oscillators for the cases of local and nonlocal coupling between the individual oscillators. It is shown that classical spiral wave and target wave regimes are observed in the network. When the nonlocal coupling between the elements is introduced, complex spatio-temporal structures, including chimera states, can be realized. We reveal and describe single- and multi-core spiral wave chimeras (SWC), as well as a novel type of chimera state which is based on target waves and called a target wave chimera (TWC). We investigate and compare the features of SWC and TWC incoherent cores. Finally, we study how the system dynamics changes when the coupling parameters are varied and discuss the impact of external noise on the observed wave structures.

Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators.

Oscillatory systems with long-range or global coupling offer promising insight into the interplay between high-dimensional (or microscopic) chaotic motion and collective interaction patterns. Within this paper, we use Lyapunov analysis to investigate whether chimera states in globally coupled Stuart-Landau (SL) oscillators exhibit collective degrees of freedom. We compare two types of chimera states, which emerge in SL ensembles with linear and nonlinear global coupling, respectively, the latter introducing a constraint that conserves the oscillation of the mean. Lyapunov spectra reveal that for both chimera states the Lyapunov exponents split into several groups with different convergence properties in the limit of large system size. Furthermore, in both cases the Lyapunov dimension is found to scale extensively and the localization properties of covariant Lypunov vectors manifest the presence of collective Lyapunov modes. Here, however, we find qualitative differences between the two types of chimera states: Whereas the ones in the system under nonlinear global coupling exhibit only slow collective modes corresponding to Lyapunov exponents equal or close to zero, those which experience the linear mean-field coupling exhibit also faster collective modes associated with Lyapunov exponents with large positive or negative values. Furthermore, for the fastest collective mode we showed that it spreads across both synchonous and incoherent oscillators.

From collective oscillation to chimera state in a nonlocally coupled excitable system

Chimera states, which consist of coexisting domains of spatially coherent and incoherent dynamics, have been widely found in nonlocally coupled oscillatory systems. We demonstrate that chimera states can emerge from excitable systems under nonlocal coupling in which isolated units only allow for the equilibrium. We theoretically reveal that nonlocal coupling-induced collective oscillation is behind the occurrence of the chimera states. We find two different types of chimera states, phase-chimera states and excitability-chimera states, depending on the coupling strength. At weak coupling strength where collective oscillation is localized around the unstable homogeneous equilibrium, the chimera states are similar to the ones in nonlocally coupled phase oscillators. For the chimera states at strong coupling strength, the dynamics of both coherent units and incoherent units shift back and forth between low-amplitude oscillation induced by collective oscillation and high-amplitude oscillation induced by excitability of local units.

Chimera at the phase-flip transition of an ensemble of identical nonlinear oscillators

A complex collective emerging behavior characterized by coexisting coherent and incoherent domains is termed as a chimera state. We bring out the existence of a new type of chimera in a nonlocally coupled ensemble of identical oscillators driven by a common dynamic environment. The latter facilitates the onset of phase-flip bifurcation/transitions among the coupled oscillators of the ensemble, while the nonlocal coupling induces a partial asynchronization among the out-of-phase synchronized oscillators at this onset. This leads to the manifestation of coexisting out-of-phase synchronized coherent domains interspersed by asynchronous incoherent domains elucidating the existence of a different type of chimera state. In addition to this, a rich variety of other collective behaviors such as clusters with phase-flip transition, conventional chimera, solitary state and complete synchronized state which have been reported using different coupling architectures are found to be induced by the employed couplings for appropriate coupling strengths. The robustness of the resulting dynamics is demonstrated in ensembles of two paradigmatic models, namely Rössler oscillators and Stuart-Landau oscillators.

Diversity of chimera-like patterns from a model of 2D arrays of neurons with nonlocal coupling

Chimera states have been studied in 1D arrays, and a variety of different chimera states have been found using different models. Research has recently been extended to 2D arrays but only to phase models of them. Here, we extend it to a nonphase model of 2D arrays of neurons and focus on the influence of nonlocal coupling. Using extensive numerical simulations, we find, surprisingly, that this system can show most types of previously observed chimera states, in contrast to previous models, where only one or a few types of chimera states can be observed in each model. We also find that this model can show some special chimera-like patterns such as gridding and multicolumn patterns, which were previously observed only in phase models. Further, we present an effective approach, i.e., removing some of the coupling links, to generate heterogeneous coupling, which results in diverse chimera-like patterns and even induces transformations from one chimera-like pattern to another.

Double-well chimeras in 2D lattice of chaotic bistable elements

We investigate spatio-temporal dynamics of a 2D ensemble of nonlocally coupled chaotic cubic maps in a bistability regime. In particular, we perform a detailed study on the transition “coherence – incoherence” for varying coupling strength for a fixed interaction radius. For the 2D ensemble we show the appearance of amplitude and phase chimera states previously reported for 1D ensembles of nonlocally coupled chaotic systems. Moreover, we uncover a novel type of chimera state, double-well chimera, which occurs due to the interplay of the bistability of the local dynamics and the 2D ensemble structure. Additionally, we find double-well chimera behavior for steady states which we call double-well chimera death. A distinguishing feature of chimera patterns observed in the lattice is that they mainly combine clusters of different chimera types: phase, amplitude and double-well chimeras.

Chimera states in networks of Van der Pol oscillators with hierarchical connectivities.

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We analyse chimera states in ring networks of Van der Pol oscillators with hierarchical coupling topology. We investigate the stepwise transition from a nonlocal to a hierarchical topology and propose the network clustering coefficient as a measure to establish a link between the existence of chimera states and the compactness of the initial base pattern of a hierarchical topology; we show that a large clustering coefficient promotes the occurrence of chimeras. Depending on the level of hierarchy and base pattern, we obtain chimera states with different numbers of incoherent domains. We investigate the chimera regimes as a function of coupling strength and nonlinearity parameter of the individual oscillators. The analysis of a network with larger base pattern resulting in larger clustering coefficient reveals two different types of chimera states and highlights the increasing role of amplitude dynamics.

Mechanisms of appearance of amplitude and phase chimera states in ensembles of nonlocally coupled chaotic systems

We explore the bifurcation transition from coherence to incoherence in ensembles of nonlocally coupled chaotic systems. It is firstly shown that two types of chimera states, namely, amplitude and phase, can be found in a network of coupled logistic maps, while only amplitude chimera states can be observed in a ring of continuous-time chaotic systems. We reveal a bifurcation mechanism by analyzing the evolution of space-time profiles and the coupling function with varying coupling coefficient and formulate the necessary and sufficient conditions for realizing the chimera states in the ensembles.

Chimera patterns under the impact of noise.

We investigate two types of chimera states, patterns consisting of coexisting spatially separated domains with coherent and incoherent dynamics, in ring networks of Stuart-Landau oscillators with symmetry-breaking coupling, under the influence of noise. Amplitude chimeras are characterized by temporally periodic dynamics throughout the whole network, but spatially incoherent behavior with respect to the amplitudes in a part of the system; they are long-living transients. Chimera death states generalize chimeras to stationary inhomogeneous patterns (oscillation death), which combine spatially coherent and incoherent domains. We analyze the impact of random perturbations, addressing the question of robustness of chimera states in the presence of white noise. We further consider the effect of symmetries applied to random initial conditions.

Rare and hidden attractors in Van der Pol-Duffing oscillators

We study the dynamics of the single and coupled van der Pol-Duffing oscillators. Each oscillator is characterized by the multistability (the coexistence of attractors). Some of the coexisting attractors have very small basins of attraction (the rare ones) and some of them do not contain equilibria in their basin of attraction (the hidden ones). We perform the detailed bifurcation analysis of these attractors and investigate how this plethora of states influences the dynamics of the network of coupled oscillators. We have observed the cluster synchronization on different attractors as well as different types of chimera states.