Nonlocally Coupled Research Articles

Cluster synchronization in a semiconductor laser

Cluster synchronization is a general phenomenon in a network of non-locally coupled oscillators. Here, we show that cluster synchronization occurs in semiconductor lasers, where the beat notes between the pairs of adjacent longitudinal modes of the laser cavity constitute a collection of coupled phase oscillators. Non-local coupling arises from the standing-wave nature of the cavity with finite mirror reflectivities, which we can actively control. Varying the coupling, we can bring the laser into a state of cluster synchronization where the two beat note families oscillate at two distinct collective frequencies. Using a coherent beat note detection technique, we show that the beat notes within the two families are synchronized in the opposite configurations—in-phase and antiphase.

Collective dynamics of nonlocally coupled Hindmarsh-Rose neurons modified by magnetic flux.

We investigate the dynamics of nonlocally coupled Hindmarsh-Rose neurons, modified by coupling the induced magnetic flux to the membrane potential with a quadratic memristor of strength k. The nonlocal coupling consists of the interaction of each neuron with its neighbors within a fixed radius, which influence the membrane potential of the neuron with coupling intensity σ. For such local dynamics and network of interactions, we investigate how variations of k and σ affect the collective dynamics. We find that when increasing k as well as when increasing σ, coherence typically increases, except for small ranges of these parameters where the opposite behavior can occur. Besides affecting coherence, varying k also affects the pattern of bursts and spikes, namely, for large enough k, burst frequency is augmented, the number and amplitude of the spikes are reduced, and quiescent periods become longer. Results are displayed for an intermediate range of interactions with radius 1/4 of the network size, but we also varied the range of interactions, ranging from first-neighbor to all-to-all couplings, observing in all cases a qualitatively similar impact of induction.

Asymmetric spiral chimeras on a spheric surface of nonlocally coupled phase oscillators.

The spiral chimera state shows a remarkable spatiotemporal pattern in a two-dimensional array of oscillators for which the coherent spiral arms coexist with incoherent cores. In this work, we report on an asymmetric spiral chimera having incoherent cores of different sizes on the spherical surface of identical phase oscillators with nonlocal coupling. This asymmetric spiral chimera exhibits a strongly symmetry-broken state in the sense that not only the coherent and incoherent domains coexist, but also their incoherent cores are nonidentical, although the underlying coupling structure is symmetric. On the basis of analyses along the Ott-Antonsen invariant manifold, the bifurcation conditions of asymmetric spiral chimeras are derived, which reveals that the asymmetric spiral chimera state emerges via a supercritical symmetry-breaking bifurcation from the symmetric spiral chimera. For the coupling function composed of two Legendre modes, rigorous stability analyses are carried out to present a complete stability diagram for different types of spiral chimeras, including the stationary symmetric and asymmetric spiral chimeras as well as breathing asymmetric spiral chimera. For the general coupling scheme the asymmetric spiral chimera occurs as a result of competition between the odd and even Legendre modes of the coupling function. Our theoretical findings are verified by using extensive numerical simulations of the model system.

Collective Dynamical Behaviors of Nonlocally Coupled Brockett Oscillators

In this study, we consider a network of nonlocally coupled Brockett oscillators (BOs) with attractive and repulsive (AR) couplings to illustrate the existence of diverse collective dynamical behaviors, whereas previous studies solely concentrated on synchronization. In the absence of coupling, the individual BO oscillator shows stable periodic oscillations (POs) or stable steady state (SS) depending on the critical values of the parameters. We first begin by examining the collective dynamics by setting the critical value of the parameters at the active (PO) region. A diverge collective dynamical states are manifested for a fixed nonlocal coupling range with rising coupling magnitude. Notably, the lower coupling strength exhibits two distinct dynamical patterns at lower and higher transients. At lesser transients, for example, transient dynamics of desynchronization, chimera, and traveling wave states are observed. At larger time periods, the transient dynamics disappear with the emergence of a synchronized state. Increasing the coupling strength results in a unique state of traveling wave or synchronized state for smaller and larger time periods depending on the coupling strength. Increasing the coupling strength further gives rise to clustering behaviors. Importantly, the considered system attains cluster oscillation death (COD) through a cluster oscillatory state (COS). Finally, there exists a chimera death at a larger coupling strength. The observed dynamical transitions are further demonstrated through the two‐parameter analysis by setting different critical thresholds.

Spiral wave chimeras in populations of oscillators coupled to a slowly varying diffusive environment

Chimera states are firstly discovered in nonlocally coupled oscillator systems. Such a nonlocal coupling arises typically as oscillators are coupled via an external environment whose characteristic time scale τ is so small (i.e., τ → o) that it could be eliminated adiabatically. Nevertheless, whether the chimera states still exist in the opposite situation (i.e., τ ≫ 1) is unknown. Here, by coupling large populations of Stuart—Landau oscillators to a diffusive environment, we demonstrate that spiral wave chimeras do exist in this oscillator-environment coupling system even when τ is very large. Various transitions such as from spiral wave chimeras to spiral waves or unstable spiral wave chimeras as functions of the system parameters are explored. A physical picture for explaining the formation of spiral wave chimeras is also provided. The existence of spiral wave chimeras is further confirmed in ensembles of FitzHugh—Nagumo oscillators with the similar oscillator-environment coupling mechanism. Our results provide an affirmative answer to the observation of spiral wave chimeras in populations of oscillators mediated via a slowly changing environment and give important hints to generate chimera patterns in both laboratory and realistic chemical or biological systems.

Bistability-induced chimeras in one-dimensional paced excitable rings with nonlocal couplings

In this paper, we provide a bistability mechanism for giving rise to a new kind of chimeras in the one-dimensional (1D) paced nonlocally coupled excitable rings without rotational coupling scheme. It is shown that the elements in the system can perform distinct modes and give rise to the chimera pattern. By analyzing the response dynamics in the corresponding local excitable model with the same pacing, the initial-excitation–dependent bistability feature is revealed as the mechanism responsible for this chimera state. Furthermore, this bistability-induced chimera state is found to be pacing dependent, and the chimera parameter regions on the phase plane of pacing amplitude and frequency are shown explicitly. Importantly, this new kind of chimera pattern can also self-organize to emerge in other paradigmatic network models, implying the genericity of the bistability mechanism in inducing chimeras in paced excitable complex networks. The present work is expected to shed light on new perspectives of the chimeras in excitable systems.

Selection of spatial modes in an ensemble of non-locally coupled chaotic maps

Purpose of this work is to determine regularities of formation of spatial structures in an ensemble of chaotic systems with non-local diffusion couplings and to study how these structures depend on the wave response of the digital filter formed by the ensemble couplings structure. Methods. The study was carried out by numerical simulation of an ensemble of logistic maps, calculation of its typical oscillatory regimes and their spectral analysis. The network was considered as a digital filter with a frequency response depending on the coupling parameters. Correlation between the spatial spectra and the amplitude-frequency response of the coupling filter and the mutual coherence of oscillations when the coupling parameters change were considered. Results. The system of couplings between chaotic maps behaves like a wave filter with selective properties, allowing spatial modes with certain wavelengths to exist and suppressing others. The selection of spatial modes is based on the wave characteristic of the coupling filter, the type of which is determined by the radius and the magnitude of couplings. At strong coupling the wave characteristics for ensembles with local and non-local couplings are qualitatively different, therefore they demonstrate essencially different behavior. Discussion. Using spectral methods for dynamics analysis systems with complex network topologies seems to be a promising approach, especially for research of synchronization and multistability in ensembles of chaotic oscillators and maps. The found regularities generalize the results known for ensembles of maps with local couplings. They also can be applied to ensembles of self-sustained oscillators.

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.

Impact of Nonlocal Interaction on Chimera States in Nonlocally Coupled Stuart–Landau Oscillators

We investigate the existence of collective dynamical states in nonlocally coupled Stuart–Landau oscillators with symmetry breaking included in the coupling term. We find that the radius of nonlocal interaction and nonisochronicity parameter play important roles in identifying the swing of synchronized states through amplitude chimera states. Collective dynamical states are distinguished with the help of strength of incoherence. Different transition routes to multi-chimera death states are analyzed with respect to the nonlocal coupling radius. In addition, we investigate the existence of collective dynamical states including traveling wave state, amplitude chimera state and multi-chimera death state by introducing higher-order nonlinear terms in the system. We also verify the robustness of the given notable properties for the coupled system.

Chimera state in a network of nonlocally coupled impact oscillators

Chimera state is a peculiar spatiotemporal pattern, wherein the coherence and incoherence coexist in the network of coupled identical oscillators. In this paper, we study the chimera states in a network of impact oscillators with nonlocal coupling. We investigate the effects of the coupling strength and the coupling range on the network behavior. The results reveal the emergence of the chimera state for significantly small values of coupling strength, and higher coupling strength values lead to unbounded motions in the oscillators. We also study the network in the case of excitation failure. We observe that the coupling helps in the maintenance of an oscillatory motion with a lower amplitude in the failed oscillator.

Travelling chimera states in systems of phase oscillators with asymmetric nonlocal coupling

We study travelling chimera states in a ring of nonlocally coupled heterogeneous (with Lorentzian distribution of natural frequencies) phase oscillators. These states are coherence-incoherence patterns moving in the lateral direction because of the broken reflection symmetry of the coupling topology. To explain the results of direct numerical simulations we consider the continuum limit of the system. In this case travelling chimera states correspond to smooth travelling wave solutions of some integro-differential equation, called the Ott–Antonsen equation, which describes the long time coarse-grained dynamics of the oscillators. Using the Lyapunov–Schmidt reduction technique we suggest a numerical approach for the continuation of these travelling waves. Moreover, we perform their linear stability analysis and show that travelling chimera states can lose their stability via fold and Hopf bifurcations. Some of the Hopf bifurcations turn out to be supercritical resulting in the observation of modulated travelling chimera states.

Chaotic maps with nonlocal coupling: Lyapunov exponents, synchronization of chaos, and characterization of chimeras

Abstract Coupled map lattices are spatially extended systems in which both space and time are discrete but allowing a continuous state variable. They have been intensively studied since they present a rich spatiotemporal dynamics, including intermittency, chimeras, and turbulence. Nonlocally coupled lattices occur in many problems of physical and biological interest, like the interaction among cells mediated by the diffusion of some chemical. In this work we investigate general features of the nonlocal coupling among maps in a regular lattice, focusing on the Lyapunov spectrum of coupled chaotic maps. This knowledge is useful for determining the stability of completely synchronized states. One of the types of nonlocal coupling investigated in this work is a smoothed finite range coupling, for which chimeras are exhibited and characterized using quantitative measures.

Spiral, Target, and Chimera Wave Structures in a Two-Dimensional Ensemble of Nonlocally Coupled van der Pol Oscillators

The dynamics of a two-dimensional ensemble of nonlocally coupled van der Pol oscillators is studied by numerical simulation. The possibility of realization of regular spiral and target waves is established. The emergence of a new type of chimera structure realized on the basis of target waves with nonlocal coupling is shown.

Traveling chimera states

In this paper we consider the Ott–Antonsen equation describing the long time coarse-grained dynamics of a ring of nonlocally coupled phase oscillators. We study traveling wave solutions relevant to traveling chimera states in the coupled oscillator system. In particular, we derive and rigorously justify asymptotic formulas for traveling wave solutions in the case of small asymmetry of nonlocal coupling. We also show that the Ott–Antonsen equation provides a reliable description of traveling chimera states for heterogeneous oscillators (with Lorentzian distribution of natural frequencies), but fails to do this for identical oscillators.

Chimera state on a spherical surface of nonlocally coupled oscillators with heterogeneous phase lags

We consider a network of coupled oscillators embedded in the surface of a sphere with nonlocal coupling strength and heterogeneous phase lags. A nonlocal coupling scheme with heterogeneous phase lags that allows the system to be solved analytically is suggested and the main effects of heterogeneity in the phase lags on the existence and stability of steady states are analyzed. We explore the stability of solutions along the Ott-Antonsen invariant manifold and present a complete bifurcation diagram for stationary patterns including the coherent, incoherent, and modulated drift states as well as chimera state. The stability analysis shows that a continuum of uniform drift states and the modulated drift state could become stable only due to the heterogeneity of the phase lags and that the chimera state is bifurcated from the modulated drift state. Our theoretical results are verified by using the direct numerical simulations of the model system.

Control of Chimera States in Multilayer Networks

Chimera states are intriguing complex spatio-temporal patterns of coexisting coherent and incoherent domains. They can often be observed in networks with nonlocal coupling topology, where each element interacts with its neighbours within a fixed range. In small-size nonlocally coupled networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. This problem can be solved with a tweezer feedback control which can stabilize and fix the position of chimera states. We analyse the action of the tweezer control in two-layer networks, where each layer is a small nonlocally coupled ring of Van der Pol oscillators. We demonstrate that tweezer control, applied to only one layer, successfully stabilizes chimera patterns in the other, uncontrolled layer, even in the case of nonidentical layers. These results might be useful for applications in multilayer networks, where one of the layers cannot be directly accessed, thus it can be effectively controlled via a neighbouring layer.

Soliton lattices in the Gross–Pitaevskii equation with nonlocal and repulsive coupling

Spatially-periodic patterns are studied in nonlocally coupled Gross–Pitaevskii equation. We show first that spatially periodic patterns appear in a model with the dipole–dipole interaction. Next, we study a model with a finite-range coupling, and show that a repulsively coupled system is closely related with an attractively coupled system and its soliton solution becomes a building block of the spatially-periodic structure. That is, the spatially-periodic structure can be interpreted as a soliton lattice. An approximate form of the soliton is given by a variational method. Furthermore, the effects of the rotating harmonic potential and spin-orbit coupling are numerically studied.

Chimera Structures in the Ensembles of Nonlocally Coupled Chaotic Oscillators

We study the structure and properties of the chimera states in the ensembles of chaotic oscillators with nonlocal coupling. It is shown that the phase and amplitude chimera states in the ensembles of chaotic oscillators with nonhyperbolic and hyperbolic attractors can be obtained using the models in the forms of two-dimensional Henon and Lozi maps. The mechanisms of birth, the structure, and the lifetime of the phase and amplitude chimeras and the regime of solitary states are studied. The chimera states in two coupled ensembles of chaotic maps are considered. The possibility of realizing a new type of the chimera structure, namely, the chimera consisting of the solitary states is demonstrated. The effects of the external and mutual synchronizations of the chimera states are described by an example of two coupled ensembles of logistic maps with nonlocal coupling. A qualitative analogy of the obtained results with the classical effect of synchronization of periodic self-sustained oscillations is discussed.

Спиральные, концентрические и химерные волновые структуры в двумерном ансамбле нелокально связанных генераторов Ван дер Поля

In the work, dynamics of a 2D ensemble of nonlocally coupled van der Pol generators are investigated using numerical simulation. The possibility of implementing regular spiral and target waves is established. The effect of the appearance of a new type of chimera structure, realized on the basis of target waves in nonlocal coupling conditions, is shown.

Synchronization of Chimera States in a Network of Many Unidirectionally Coupled Layers of Discrete Maps

We study numerically external synchronization of chimera states in a network of many unidirectionally coupled layers, each representing a ring of nonlocally coupled discretetime systems. The dynamics of each element in the network is described by either the logistic map or the bistable cubic map. We consider two cases: when all M unidirectionally coupled layers are identical and when (M − 1) identical layers differ from the first driving layer in their nonlocal coupling parameters. It is shown that the master chimera state in the first layer can be retranslating along the network with small distortions which are defined by a parameter mismatch. We also analyze the dependence of the mean-square deviation of the structure in the ith layer on the nonlocal coupling parameters.