Phase-flip chimera induced by environmental nonlocal coupling

Chimera states and cluster solutions in Hindmarsh-Rose neural networks with state resetting process

The neuronal state resetting model is a hybrid system, which combines neuronal system with state resetting process. As the membrane potential reaches a certain threshold, the membrane potential and recovery current are reset. Through the resetting process, the neuronal system can produce abundant new firing patterns. By integrating with the state resetting process, the neuronal system can generate irregular limit cycles (limit cycles with impulsive breakpoints), resulting in repetitive spiking or bursting with firing peaks which can not exceed a presetting threshold. Although some studies have discussed the state resetting process in neurons, it has not been addressed in neural networks so far. In this paper, we consider chimera states and cluster solutions in Hindmarsh-Rose neural networks with state resetting process. The network structures are based on regular ring structures and the connections among neurons are assumed to be bidirectional. Chimera and cluster states are two types of phenomena related to synchronization. For neural networks, the chimera state is a self-organization phenomenon in which some neuronal nodes are synchronous while the others are asynchronous. Cluster synchronization divides the system into several subgroups based on their synchronization characteristics, with neuronal nodes in each subgroup being synchronous. By improving previous chimera measures, we detect the spike inspire time instead of the state variable and calculate the time between two adjacent spikes. We then discuss the incoherence, chimera state, and coherence of the constructed neural networks using phase diagrams, time series diagrams, and probability density histograms. Besides, we further contrast the cluster solutions of the system under local and global coupling, respectively. The subordinate state resetting process enriches the firing mode of the proposed Hindmarsh-Rose neural networks.

Frequency chimera state induced by differing dynamical timescales

We report the occurrence of a self-emerging frequency chimera state in spatially extended systems of coupled oscillators, where the coherence and incoherence are defined with respect to the emergent frequency of the oscillations. This is generated by the local coupling among nonlinear oscillators evolving under differing dynamical timescales starting from random initial conditions. We show how they self-organize to structured patterns with spatial domains of coherence that are in frequency synchronization, coexisting with domains that are incoherent in frequencies. Our study has relevance in understanding such patterns observed in real-world systems like neuronal systems, power grids, social and ecological networks, where differing dynamical timescales is natural and realistic among the interacting systems.

Symmetry breakings in two populations of oscillators coupled via diffusive environments: Chimera and heterosynchrony

We consider two diffusively coupled populations of identical oscillators, where the oscillators in each population are coupled with a common dynamic environment. Existence and stability of a variety of stationary states are analyzed on the basis of the Ott-Antonsen reduction method, which reveals that the chimera state occurs under the diffusive coupling scheme. Furthermore, we find an exotic symmetry-breaking behavior, the so-called the heterosynchronous state, in which each population exhibits in-phase coherence, while the order parameters of two populations rotate at different phase velocities. The chimera and heterosynchronous states emerge from bistabilities of distinct states for decoupled population and occur as a unique continuation for weak diffusive couplings. The heterosynchronous state is caused by an indirect coupling scheme via dynamic environments and could occur for a finite-size system as well, even for the system that consists of one oscillator per population.

Amplitude chimera and chimera death induced by external agents in two-layer networks

We report the emergence of stable amplitude chimeras and chimera death in a two-layer network where one layer has an ensemble of identical nonlinear oscillators interacting directly through local coupling and indirectly through dynamic agents that form the second layer. The nonlocality in the interaction among the dynamic agents in the second layer induces different types of chimera-related dynamical states in the first layer. The amplitude chimeras developed in them are found to be extremely stable, while chimera death states are prevalent for increased coupling strengths. The results presented are for a system of coupled Stuart-Landau oscillators and can, in general, represent systems with short-range interactions coupled to another set of systems with long-range interactions. In this case, by tuning the range of interactions among the oscillators or the coupling strength between two types of systems, we can control the nature of chimera states and the system can also be restored to homogeneous steady states. The dynamic agents interacting nonlocally with long-range interactions can be considered as a dynamic environment or a medium interacting with the system. We indicate how the second layer can act as a reinforcement mechanism on the first layer under various possible interactions for desirable effects.

Distinct collective states due to trade-off between attractive and repulsive couplings

We investigate the effect of repulsive coupling together with an attractive coupling in a network of nonlocally coupled oscillators. To understand the complex interaction between these two couplings we introduce a control parameter in the repulsive coupling which plays a crucial role in inducing distinct complex collective patterns. In particular, we show the emergence of various cluster chimera death states through a dynamically distinct transition route, namely the oscillatory cluster state and coherent oscillation death state as a function of the repulsive coupling in the presence of the attractive coupling. In the oscillatory cluster state, the oscillators in the network are grouped into two distinct dynamical states of homogeneous and inhomogeneous oscillatory states. Further, the network of coupled oscillators follow the same transition route in the entire coupling range. Depending upon distinct coupling ranges, the system displays different number of clusters in the death state and oscillatory state. We also observe that the number of coherent domains in the oscillatory cluster state exponentially decreases with increase in coupling range and obeys a power-law decay. Additionally, we show analytical stability for observed solitary state, synchronized state, and incoherent oscillation death state.

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

We investigate the occurrence of collective dynamical states such as transient amplitude chimera, stable amplitude chimera, and imperfect breathing chimera states in a locally coupled network of Stuart-Landau oscillators. In an imperfect breathing chimera state, the synchronized group of oscillators exhibits oscillations with large amplitudes, while the desynchronized group of oscillators oscillates with small amplitudes, and this behavior of coexistence of synchronized and desynchronized oscillations fluctuates with time. Then, we analyze the stability of the amplitude chimera states under various circumstances, including variations in system parameters and coupling strength, and perturbations in the initial states of the oscillators. For an increase in the value of the system parameter, namely, the nonisochronicity parameter, the transient chimera state becomes a stable chimera state for a sufficiently large value of coupling strength. In addition, we also analyze the stability of these states by perturbing the initial states of the oscillators. We find that while a small perturbation allows one to perturb a large number of oscillators resulting in a stable amplitude chimera state, a large perturbation allows one to perturb a small number of oscillators to get a stable amplitude chimera state. We also find the stability of the transient and stable amplitude chimera states and traveling wave states for an appropriate number of oscillators using Floquet theory. In addition, we also find the stability of the incoherent oscillation death states.

Chimeralike states in networks of bistable time-delayed feedback oscillators coupled via the mean field

We study the collective dynamics of oscillators in a network of identical bistable time-delayed feedback systems globally coupled via the mean field. The influence of delay and inertial properties of the mean field on the collective behavior of globally coupled oscillators is investigated. A variety of oscillation regimes in the network results from the presence of bistable states with substantially different frequencies in coupled oscillators. In the physical experiment and numerical simulation we demonstrate the existence of chimeralike states, in which some of the oscillators in the network exhibit synchronous oscillations, while all other oscillators remain asynchronous.

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.

Occurrence and stability of chimera states in coupled externally excited oscillators

We studied the phenomenon of chimera states in networks of non-locally coupled externally excited oscillators. Units of the considered networks are bi-stable, having two co-existing attractors of different types (chaotic and periodic). The occurrence of chimeras is discussed, and the influence of coupling radius and coupling strength on their co-existence is analyzed (including typical bifurcation scenarios). We present a statistical analysis and investigate sensitivity of the probability of observing chimeras to the initial conditions and parameter values. Due to the fact that each unit of the considered networks is individually excited, we study the influence of the excitation failure on stability of observed states. Typical transitions are shown, and changes in network's dynamics are discussed. We analyze systems of coupled van der Pol-Duffing oscillators and the Duffing ones. Described chimera states are robust as they are observed in the wide regions of parameter values, as well as in other networks of coupled forced oscillators.