Nonlocally Coupled Research Articles

Multiscale interaction promotes chimera states in complex networks

We report emergence of chimera states in a multiscale network that defines a network of interconnected networks, a network of a global ring of nodes whose each node, in turn, is a member of an individual subnetwork of another ring of nodes. We reveal a variety of spatiotemporal dynamics which ensues in such complex networks by varying the inner topology of the nonlocally coupled subnetworks from local to nonlocal coupling. We find that an increment in the local connectivity of the subnetworks enhances the span of the chimera region in parameter space. We consider the Kuramoto–Sakaguchi phase model to represent each node of the network and our numerical findings show that the topology of the subnetworks greatly influences the emergence of chimera states in the global ring. We also perform an analytical study on the phase oscillator based network using the Ott–Antonsen approach, and the analytical results are found to be consistent with the numerical outcomes. Furthermore, due to high relevance of the proposed network architecture to the human brain, we consider a more realistic network of Hindmarsh–Rose neuron model and reveal a similar phenomenon when the neurons in each of the subnetworks communicate via chemical synapses while information among the subnetworks passes through gap junctions.

Mechanism of solitary state appearance in an ensemble of nonlocally coupled Lozi maps

We study the peculiarities of the solitary state appearance in the ensemble of nonlocally coupled chaotic maps. We show that nonlocal coupling and features of the partial elements lead to arising of multistability in the system. The existence of solitary state is caused by formation of two attractive sets with different basins of attraction. Their basins are analysed depending on coupling parameters.

Mechanism of realizing a solitary state chimera in a ring of nonlocally coupled chaotic maps

We study the mechanism of appearance of a solitary state chimera in a ring of nonlocally coupled two-dimensional chaotic maps. We show that both solitary states and the solitary state chimera can be induced by noise which multiplicatively modulates the nonlocal coupling coefficient in the ensemble of nonlocally coupled Henon maps. It is demonstrated that these states can also arise due to multistability in the network. It has been firstly shown and explained the possibility of obtaining the solitary states and the solitary state chimera in the one-dimensional ring of nonlocally coupled Henon maps. Basins of attraction of coexisting attractors essentially depend on the spatiotemporal structure, which is realized in the ring, as well as on the position of a particular oscillator among the other oscillators. We have found out that the riddling effect of the basins of attraction can be observed when the network state shows an amplitude chimera.

Stable amplitude chimera in a network of coupled Stuart-Landau oscillators

In many natural systems, attractive coupling together with repulsive coupling plays a vital role in determining their evolutionary dynamics. We investigate the stabilization of amplitude chimera through repulsive coupling in the presence of attractive coupling in a system of nonlocally coupled oscillators. The nonlocal repulsive coupling can facilitate the emergence of stable amplitude chimera even for random initial conditions contrasting with the earlier investigations, where the amplitude chimera was observed just as a transient state and that too for a specific cluster initial conditions. The stability of the observed amplitude chimera is analyzed using Floquet theory. To elucidate the transition among the distinct dynamical states, we find the average number of inhomogeneous oscillators as a function of the coupling strength and show that the transition among the dynamical states exhibits hysteresis. Further, we deduce analytically the critical stability curve that separates the oscillatory (amplitude chimera and traveling wave) states from the death (multi-incoherent oscillation death, cluster chimera death, cluster oscillation death) states. We also analyze the influence of the nonisochronicity parameter and noise on the stable amplitude chimera. We report that the nonisochronicity parameter favors the traveling wave state from incoherent death through the stable amplitude chimera state.

Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution

Twisted states, referred to traveling waves in an array of nonlocally coupled phase oscillators, have drawn some attention in recent years. In this work, we study a one-dimensional ring of phase oscillators with nonlocal coupling and a bimodal natural frequency distribution. We show that twisted standing waves and stationary twisted states appear successively with the increase of the coupling strength. In the continuum limit, we derive a low-dimensional reduced equation using the Ott–Antonsen ansatz, which verifies the twisted states in the simulations of finite networks of oscillators. We also theoretically investigate the stationary twisted states and their stabilities by using the reduced equation.

Symmetry-broken coherent state in a ring of nonlocally coupled identical oscillators.

We report on a modulated coherent state in a ring of nonlocally coupled oscillators. Although the identical oscillators are all synchronized under the symmetric coupling, the phase configuration has an inhomogeneous structure. This symmetry-broken coherent state exists only for a nonlocal coupling with both attracting and repulsive interactions, depending on the distance between oscillators, and emerges via a continuous bifurcation from a uniformly coherent state. We analyze the existence and stability of the modulated coherent states on the basis of Ott-Antonsen equations for the local order parameter. Our theoretical results are verified using the numerical simulations.

Enhancing coherence via tuning coupling range in nonlocally coupled Stuart\u2013Landau oscillators

Nonlocal coupling, as an important connection topology among nonlinear oscillators, has attracted increasing attention recently with the research boom of chimera states. So far, most previous investigations have focused on nonlocally coupled systems interacted via similar variables. In this work, we report the evolutions of dynamical behaviors in the nonlocally coupled Stuart–Landau oscillators by applying conjugate variables feedback. Through rigorous analysis, we find that the oscillation death (OD) can convert into the amplitude death (AD) via the cluster state with the increasing of coupling range, making the AD regions to be expanded infinitely along two directions of both the natural frequency and the coupling strength. Moreover, the limit cycle oscillation (OS) region and the mixed region of OD and OS will turn to anti-synchronization state through amplitude-mediated chimera. Therefore, the procedure from local coupling to nonlocal one implies indeed the continuous enhancement of coherence among neighboring oscillators in coupled systems.

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.

Chimera and modulated drift states in a ring of nonlocally coupled oscillators with heterogeneous phase lags

We consider a ring of phase oscillators with nonlocal coupling strength and heterogeneous phase lags. We analyze the effects of heterogeneity in the phase lags on the existence and stability of a variety of steady states. A nonlocal coupling with heterogeneous phase lags that allows the system to be solved analytically is suggested and the stability of solutions along the Ott-Antonsen invariant manifold is explored. We present a complete bifurcation diagram for stationary patterns including the uniform drift and modulated drift states as well as chimera state, which reveals that the stable modulated drift state and a continuum of metastable drift states could occur due to the heterogeneity of the phase lags. We verify our theoretical results using the direct numerical simulations of the model system.

Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions.

The dynamics of a large array of coupled semiconductor lasers is studied numerically for a nonlocal coupling scheme. Our focus is on chimera states, a self-organized spatiotemporal pattern of coexisting coherence and incoherence. In laser systems, such states have been previously found for global and nearest-neighbor coupling, mainly in small networks. The technological advantage of large arrays has motivated us to study a system of 200 nonlocally coupled lasers with respect to the emerging collective dynamics. Moreover, the nonlocal nature of the coupling allows us to obtain robust chimera states with multiple (in)coherent domains. The crucial parameters are the coupling strength, the coupling phase and the range of the nonlocal interaction. We find that multiclustered chimera states exist in a wide region of the parameter space and we provide quantitative characterization for the obtained spatiotemporal patterns. By proposing two different experimental setups for the realization of the nonlocal coupling scheme, we are confident that our results can be confirmed in the laboratory.

Chimera states and the interplay between initial conditions and non-local coupling.

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We study chimera states in a network of non-locally coupled Stuart-Landau oscillators. We investigate the impact of initial conditions in combination with non-local coupling. Based on an analytical argument, we show how the coupling phase and the coupling strength are linked to the occurrence of chimera states, flipped profiles of the mean phase velocity, and the transition from a phase- to an amplitude-mediated chimera state.

“Coherence–incoherence” transition in ensembles of nonlocally coupled chaotic oscillators with nonhyperbolic and hyperbolic attractors

We consider in detail similarities and differences of the “coherence–incoherence” transition in ensembles of nonlocally coupled chaotic discrete-time systems with nonhyperbolic and hyperbolic attractors. As basic models we employ the Henon map and the Lozi map. We show that phase and amplitude chimera states appear in a ring of coupled Henon maps, while no chimeras are observed in an ensemble of coupled Lozi maps. In the latter, the transition to spatio-temporal chaos occurs via solitary states. We present numerical results for the coupling function which describes the impact of neighboring oscillators on each partial element of an ensemble with nonlocal coupling. Varying the coupling strength we analyze the evolution of the coupling function and discuss in detail its role in the “coherence–incoherence” transition in the ensembles of Henon and Lozi maps.

Chimeralike states in two distinct groups of identical populations of coupled Stuart-Landau oscillators.

We show the existence of chimeralike states in two distinct groups of identical populations of globally coupled Stuart-Landau oscillators. The existence of chimeralike states occurs only for a small range of frequency difference between the two populations, and these states disappear for an increase of mismatch between the frequencies. Here the chimeralike states are characterized by the synchronized oscillations in one population and desynchronized oscillations in another population. We also find that such states observed in two distinct groups of identical populations of nonlocally coupled oscillators are different from the above case in which coexisting domains of synchronized and desynchronized oscillations are observed in one population and the second population exhibits synchronized oscillations for spatially prepared initial conditions. Perturbation from such spatially prepared initial condition leads to the existence of imperfectly synchronized states. An imperfectly synchronized state represents the existence of solitary oscillators which escape from the synchronized group in population I and synchronized oscillations in population II. Also the existence of chimera state is independent of the increase of frequency mismatch between the populations. We also find the coexistence of different dynamical states with respect to different initial conditions, which causes multistability in the globally coupled system. In the case of nonlocal coupling, the system does not show multistability except in the cluster state region.

Phase-flip chimera induced by environmental nonlocal coupling.

We report the emergence of a collective dynamical state, namely, the phase-flip chimera, from an ensemble of identical nonlinear oscillators that are coupled indirectly via the dynamical variables from a common environment, which in turn are nonlocally coupled. The phase-flip chimera is characterized by the coexistence of two adjacent out-of-phase synchronized coherent domains interspersed by an incoherent domain, in which the nearby oscillators are in out-of-phase synchronized states. Attractors of the coherent domains are either from the same or from different basins of attractions, depending on whether they are periodic or chaotic. The conventional chimera precedes the phase-flip chimera in general. Further, the phase-flip chimera emerges after the completely synchronized evolution of the ensemble, in contrast to conventional chimeras, which emerge as an intermediate state between completely incoherent and coherent states. We have also characterized the observed dynamical transitions using the strength of incoherence, probability distribution of the correlation coefficient, and framework of the master stability function.

Different kinds of chimera death states in nonlocally coupled oscillators.

We investigate the significance of nonisochronicity parameter in a network of nonlocally coupled Stuart-Landau oscillators with symmetry breaking form. We observe that the presence of nonisochronicity parameter leads to structural changes in the chimera death region while varying the strength of the interaction. This gives rise to the existence of different types of chimera death states such as multichimera death state, type I periodic chimera death (PCD) state, and type II periodic chimera death state. We also find that the number of periodic domains in both types of PCD states decreases exponentially with an increase of coupling range and obeys a power law under nonlocal coupling. Additionally, we also analyze the structural changes of chimera death states by reducing the system of dynamical equations to a phase model through the phase reduction. We also briefly study the role of nonisochronicity parameter on chimera states, where the existence of a multichimera state with respect to the coupling range is pointed out. Moreover, we also analyze the robustness of the chimera death state to perturbations in the natural frequencies of the oscillators.

Phase oscillators in modular networks: The effect of nonlocal coupling.

We study the dynamics of nonlocally coupled phase oscillators in a modular network. The interactions include a phase lag, α. Depending on the various parameters the system exhibits a number of different dynamical states. In addition to global synchrony there can also be modular synchrony when each module can synchronize separately to a different frequency. There can also be multicluster frequency chimeras, namely coherent domains consisting of modules that are separately synchronized to different frequencies, coexisting with modules within which the dynamics is desynchronized. We apply the Ott-Antonsen ansatz in order to reduce the effective dimensionality and thereby carry out a detailed analysis of the different dynamical states.

Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities.

We study numerically the development of chimera states in networks of nonlocally coupled oscillators whose limit cycles emerge from a Hopf bifurcation. This dynamical system is inspired from population dynamics and consists of three interacting species in cyclic reactions. The complexity of the dynamics arises from the presence of a limit cycle and four fixed points. When the bifurcation parameter increases away from the Hopf bifurcation the trajectory approaches the heteroclinic invariant manifolds of the fixed points producing spikes, followed by long resting periods. We observe chimera states in this spiking regime as a coexistence of coherence (synchronization) and incoherence (desynchronization) in a one-dimensional ring with nonlocal coupling and demonstrate that their multiplicity depends on both the system and the coupling parameters. We also show that hierarchical (fractal) coupling topologies induce traveling multichimera states. The speed of motion of the coherent and incoherent parts along the ring is computed through the Fourier spectra of the corresponding dynamics.

Mean-field-diffusion–induced chimera death state

Recently a novel dynamical state, called the chimera death, has been discovered in a network of nonlocally coupled identical oscillators (Zakharova A., Kapeller M. and Schöll E., Phys. Rev. Lett., 112 (2014) 154101), which is defined as the coexistence of spatially coherent and incoherent oscillation death state. This state arises due to the interplay of nonlocality and symmetry breaking and thus it 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 oscillators with mean-field diffusive coupling and thus we establish that a nonlocal coupling is not essential to obtain chimera death. We identify a new transition route to the chimera death state, namely the transition from in-phase synchronized oscillation to chimera death via global amplitude death state. We ascribe the occurrence of chimera death to the bifurcation structure of the network in the limiting condition and show that multi-cluster chimera death states can be achieved by a proper choice of initial conditions.

Controlling chimeras

Coupled phase oscillators model a variety of dynamical phenomena in nature and technological applications. Non-local coupling gives rise to chimera states which are characterized by a distinct part of phase-synchronized oscillators while the remaining ones move incoherently. Here, we apply the idea of control to chimera states: using gradient dynamics to exploit drift of a chimera, it will attain any desired target position. Through control, chimera states become functionally relevant; for example, the controlled position of localized synchrony may encode information and perform computations. Since functional aspects are crucial in (neuro-)biology and technology, the localized synchronization of a chimera state becomes accessible to develop novel applications. Based on gradient dynamics, our control strategy applies to any suitable observable and can be generalized to arbitrary dimensions. Thus, the applicability of chimera control goes beyond chimera states in non-locally coupled systems.