Chimera Death Research Articles

Coupling dynamics of locally active memristor based neurons.

Brain-like dynamics require third-order or higher-order complexity. In order to investigate the coupling neuromorphic behaviors of identical third-order memristive neurons, this paper begins with the aim of exploring two identical neuron based dynamics under distinct operating regimes and coupling strengths. Without coupling, the single neuron can exhibit resting states, periodic spikes, or chaos depending on the bias condition. The uncoupled resting neurons can be activated by resistive coupling, inducing inhomogeneous resting states (static Smale paradox) and inhomogeneous spikes (dynamic Smale paradox) due to the edge of chaos regime. Considering the single neuron at the periodic spikes or chaotic states, the coupled neurons can mimic shocking oscillation death, non-periodic asynchronization, and periodic synchronization via the Hopf bifurcation theory. From the above analyses, an artificial ring neural network is constructed using 100 memristive neurons and resistive synapses to further study the coupled mechanism, generating exotic spatiotemporal patterns such as chimera death, amplitude chimera, solitary states, and asynchronization because of symmetry breaking. This sheds new light on exploring exotic spatiotemporal patterns of networks based on memristive neurons from the perspective of the nonlinear circuit theory.

Chimera states in fractional-order coupled Rayleigh oscillators

Chimera states, characterized by the coexistence of synchronized and desynchronized oscillators, have been extensively described via integer differential equations over the past few decades. However, it is still unclear whether and under which conditions chimera states will appear in terms of the coupled systems involving fractional derivatives. In this work, we investigate the chimera states of Rayleigh oscillators in the fractional-order systems with attractive and repulsive couplings. Firstly, we introduce the repulsive control factor and show the various transition routes from the desynchronized state to the incoherent oscillation death by altering this factor and the fractional-order derivatives. Then, it is demonstrated that as the coupling strength varies, a vast repertoire of interesting collective behaviors will emerge, such as synchronization, solitary state, traveling wave, traveling chimera, oscillatory cluster, imperfect traveling chimera state, imperfect amplitude chimera state, and chimera death. Finally, we discuss the effect of attractive and repulsive couplings and find that decreasing the repulsive control factor can reduce the repulsive interaction, allowing the attractive coupling to dominate and promote the emergence of the coherent state.

Amplitude chimeras and bump states with and without frequency entanglement: a toy model

When chaotic oscillators are coupled in complex networks a number of interesting synchronization phenomena emerge. Notable examples are the frequency and amplitude chimeras, chimera death states, solitary states as well as combinations of these. In a previous study (Provata 2020 J. Phys. Complex. 1 025006), a toy model was introduced addressing possible mechanisms behind the formation of frequency chimera states. In the present study a variation of the toy model is proposed to address the formation of amplitude chimeras. The proposed oscillatory model is now equipped with an additional 3rd order equation modulating the amplitude of the network oscillators. This way, the single oscillators are constructed as bistable in amplitude and depending on the initial conditions their amplitude may result in one of the two stable fixed points. Numerical simulations demonstrate that when these oscillators are nonlocally coupled in networks, they organize in domains with alternating amplitudes (related to the two fixed points), naturally forming amplitude chimeras. A second extension of this model incorporates nonlinear terms merging amplitude together with frequency, and this extension allows for the spontaneous production of composite amplitude-and-frequency chimeras occurring simultaneously in the network. Moreover the extended model allows to understand the emergence of bump states via the continuous passage from chimera states, when both fixed point amplitudes are positive, to bump states when one of the two fixed points vanishes. The synchronization properties of the network are studied as a function of the system parameters for the case of amplitude chimeras, bump states and composite amplitude-and-frequency chimeras. The proposed mechanisms of creating domains with variable amplitudes and/or frequencies provide a generic scenario for understanding the formation of the complex synchronization phenomena observed in networks of coupled nonlinear and chaotic oscillators.

Effect of fractional derivatives on amplitude chimeras and symmetry-breaking death states in networks of limit-cycle oscillators.

We study networks of coupled oscillators whose local dynamics are governed by the fractional-order versions of the paradigmatic van der Pol and Rayleigh oscillators. We show that the networks exhibit diverse amplitude chimeras and oscillation death patterns. The occurrence of amplitude chimeras in a network of van der Pol oscillators is observed for the first time. A form of amplitude chimera, namely, "damped amplitude chimera" is observed and characterized, where the size of the incoherent region(s) increases continuously in the course of time, and the oscillations of drifting units are damped continuously until they are quenched to steady state. It is found that as the order of the fractional derivative decreases, the lifetime of classical amplitude chimeras increases, and there is a critical point at which there is a transition to damped amplitude chimeras. Overall, a decrease in the order of fractional derivatives reduces the propensity to synchronization and promotes oscillation death phenomena including solitary oscillation death and chimera death patterns that were unobserved in networks of integer-order oscillators. This effect of the fractional derivatives is verified by the stability analysis based on the properties of the master stability function of some collective dynamical states calculated from the block-diagonalized variational equations of the coupled systems. The present study generalizes the results of our recently studied network of fractional-order Stuart-Landau oscillators.

Chimera patterns with spatial random swings between periodic attractors in a network of FitzHugh-Nagumo oscillators.

For the study of symmetry-breaking phenomena in neuronal networks, simplified versions of the FitzHugh-Nagumo model are widely used. In this paper, these phenomena are investigated in a network of FitzHugh-Nagumo oscillators taken in the form of the original model and it is found that it exhibits diverse partial synchronization patterns that are unobserved in the networks with simplified models. Apart from the classical chimera, we report a new type of chimera pattern whose incoherent clusters are characterized by spatial random swings among a few fixed periodic attractors. Another peculiar hybrid state is found that combines the features of this chimera state and a solitary state such that the main coherent cluster is interspersed with some nodes with identical solitary dynamics. In addition, oscillation death including chimera death emerges in this network. A reduced model of the network is derived to study oscillation death, which helps explaining the transition from spatial chaos to oscillation death via the chimera state with a solitary state. This study deepens our understanding of chimera patterns in neuronal networks.

Multistable chimera states in a smallest population of three coupled oscillators.

We uncover the emergence of distinct sets of multistable chimera states in addition to chimera death and synchronized states in a smallest population of three globally coupled oscillators with mean-field diffusive coupling. Sequence of torus bifurcations result in the manifestation of distinct periodic orbits as a function of the coupling strength, which in turn result in the genesis of distinct chimera states constituted by two synchronized oscillators coexisting with an asynchronous oscillator. Two subsequent Hopf bifurcations result in homogeneous and inhomogeneous steady states resulting in desynchronized steady states and chimera death state among the coupled oscillators. The periodic orbits and the steady states lose their stability via a sequence of saddle-loop and saddle-node bifurcations finally resulting in a stable synchronized state. We have generalized these results to N coupled oscillators and also deduced the variational equationscorresponding to the perturbation transverse to the synchronization manifold and corroborated the synchronized state in the two-parameter phase diagrams using its largest eigenvalue. Chimera states in three coupled oscillators emerge as a solitary state in N coupled oscillator ensemble.

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.

Diverse chimera and symmetry-breaking patterns induced by fractional derivation effect in a network of Stuart-Landau oscillators

In this paper, we study the impact of fractional derivation on the symmetry-breaking dynamics of a network of coupled fractional-order Stuart-Landau oscillators. It is found that the value of the fractional derivatives order has a significant impact on the network dynamics. A decrease in the derivatives order results in complex dynamical patterns that are unobserved in the coupled integer-order system, namely solitary oscillation death state, multi-cluster chimera death, incoherent oscillation death, one-headed amplitude chimera, two-headed amplitude chimera, coherent travelling waves, and amplitude chimera on traveling waves background. It is also found that the fractional derivation can contribute to extend the lifetime of the amplitude chimeras. The stability of some of these states is investigated analytically, which allows to predict their regions of existence in the parameter space. Then, these dynamical features of the network are characterized to be mapped in the global parameter space. Significantly, with a decreasing derivatives order, the amplitude chimeras whose drifting oscillators are characterized by damped oscillations become short lived transient states that are eventually transformed into incoherent oscillation death states. A lower value of the fractional derivatives order inhibits amplitude chimeras and in-phase synchronized state, and promotes diverse oscillation death and chimera death states. The present study will deepen our understanding of chimera states in several natural systems that are essentially fractional in their dynamics.

Symmetry breaking by power-law coupling

• Symmetry breaking dynamics in a network of Rayleigh oscillators are explored. • Power-law type interaction induces amplitude chimera and oscillation death. • A novel transient behavior where amplitude chimeras and oscillation death states coexist is reported. • The occurrence of amplitude chimera is established using the theory of Floquet multipliers. We study the symmetry breaking phenomena in a network of Rayleigh oscillators coupled through power-law coupling whose interaction range is controlled by a power-law exponent. We show that in a broad range of the power-law exponent several symmetry-breaking states, such as amplitude chimeras and oscillation death are induced. Further, we observe an interesting transient behavior where amplitude chimeras and oscillation death states coexist. We establish the occurrence of the amplitude chimera using the theory of Floquet multipliers as well as other correlation measures. This paper deepens our understanding of the impact of power-law type interaction on symmetry-breaking states in a network of coupled oscillators.

Trade-off between filtering and symmetry breaking mean-field coupling in inducing macroscopic dynamical states

We study the manifestation of the competing interaction between the mean-field intensity and the symmetry breaking coupling on the phenomenon of aging transition in an ensemble of limit-cycle oscillators comprising of active and inactive oscillators. Further, we also introduce filtering in both the intrinsic and extrinsic variables of the mean-field diffusive coupling to investigate the counter-intuitive effect of both filterings. We find that large values of the mean-field intensity near unity favor the oscillatory nature of the ensemble, whereas low values favor the onset of the aging transition and heterogeneous dynamical states such as cluster oscillation death and chimera death states even at low values of the symmetry breaking coupling strength. Heterogeneous dynamical states predominates at large values of the coupling strength in all available parameter spaces. We also uncover that even a weak intrinsic filtering favors the aging transition and heterogeneous dynamical states, while a feeble extrinsic filtering favors the oscillatory state. Chimera death state is observed among the active oscillators for the first time in the aging literature. Our results can lead to engineering the dynamical states as desired by an appropriate choice of the control parameters. Further, the transition from the oscillatory to the aging state occurs via an inverse Hopf bifurcation, while the transition from the aging state to the cluster oscillation death states emerges through a supercritical pitch-fork bifurcation. The deduced analytical bifurcation curves are in good agreement with the numerical boundaries of the observed dynamical states.

Symmetry breaking-induced state-dependent aging and chimera-like death state

Aging transition refers to the manifestation of the homogeneous steady state beyond a critical ratio of the inactive oscillators in a network of active and inactive oscillators. In contrast, symmetry breaking in coupled dynamical systems always results in heterogeneous states. In this work, surprisingly, we find that the symmetry breaking coupling facilitates the onset of the homogeneous steady state among the population at the critical proportion of the inactive oscillators despite the presence of a large number of the active oscillators. Further, increase in the natural frequency of the oscillation and the number of the inactive oscillators are conducive to the onset of the aging transition. Interestingly, chimera-like death state is observed in the study related to the aging transition for the first time in the literature in addition to the actual chimera death states among the active oscillator for the random initial conditions. The critical curves corresponding to the stability of the aging transition and the chimera-like death states are deduced, which agrees perfectly with the simulation results. Even a feeble decrease in the feedback factor in the coupling destabilizes the stable aging state thereby facilitating the oscillatory state in the entire parameter space despite the presence of a large proportion of the inactive oscillators increasing the robustness of the network against disorders.

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.

Inhomogeneous to homogeneous dynamical states through symmetry breaking dynamics

We explore the dynamical transitions facilitated by the symmetry breaking dynamical states in a directly coupled limit-cycle oscillators, when the oscillators are interacting through attractive and repulsive couplings. Initially, we show that the transition from out-of-phase synchronized (OPS) state to in-phase synchronized (IPS) state is mediated by the onset of symmetry breaking oscillatory (SBO) state using two coupled systems. Similarly, we find the emergence of symmetry breaking oscillation death (AOD) state when the system transits from oscillation death (OD) state to nontrivial amplitude death (NAD) state. Further, we deduce the analytical stability conditions for all the observed dynamical states, namely the IPS, OPS, NAD, OD, including SBO and AOD states. Additionally, the investigation on symmetry breaking dynamics in a ring of unidirectionally coupled oscillators reveals that the breaking of symmetry leads to the traveling chimera (TC) and symmetry breaking incoherent oscillation death states. We also report that the transient nature of TC state obeys a power-law relation near the stable TC region as a function of the coupling strength. From the results, it is evident that the trade-off between the attractive and the repulsive couplings induces the symmetry breaking dynamics in the entire range of the system frequency.

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.

Asymmetry in initial cluster size favors symmetry in a network of oscillators.

Counterintuitive to the common notion of symmetry breaking, asymmetry favors synchrony in a network of oscillators. Our observations on an ensemble of identical Stuart-Landau systems under a symmetry breaking coupling support our conjecture. As usual, for a complete deterministic and the symmetric choice of initial clusters, a variety of asymptotic states, namely, multicluster oscillation death (1-OD, 3-OD, and -OD), chimera states, and traveling waves emerge. Alternatively, multiple chimera death (1-CD, 3-CD, and -CD) and completely synchronous states emerge in the network whenever some randomness is added to the symmetric initial states. However, in both the cases, an increasing asymmetry in the initial cluster size restores symmetry in the network, leading to the most favorable complete synchronization state for a broad range of coupling parameters. We are able to reduce the network model using the mean-field approximation that reproduces the dynamical features of the original network.

Stable amplitude chimera states and chimera death in repulsively coupled chaotic oscillators

Amplitude chimera states, representing a spontaneous symmetry breaking of a population of coupled identical oscillators into two distinct clusters with one oscillating in spatial coherent amplitude, while the other displaying oscillations in a spatially incoherent manner, have been observed as a kind of transient dynamics in the process of transition to the in-phase synchronization in coupled limit-cycle oscillators. Here, we obtain a kind of stable amplitude chimera state in the chaotic regime of a system of repulsively coupled Lorenz oscillators. With the increment of the coupling strength, the coupled oscillators transit from spatiotemporal chaos to amplitude chimera states then to coherent oscillation death or chimera death states. Moreover, the number of clusters in amplitude chimera patterns has a power-law dependence on the number of coupled neighbors. The amplitude chimera and the chimera death states coexist at certain coupling strength. Moreover, the amplitude chimera and the amplitude death patterns are related to the initial condition for given coupling strength. Our findings of amplitude chimera states and chimera death states in coupled chaotic system may enrich the knowledge of the symmetry-breaking-induced pattern formation.

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.

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.

Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics

We review chimera patterns, which consist of coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics in networks of identical oscillators. We focus on chimera states involving amplitude as well as phase dynamics, complex topologies like small-world or hierarchical (fractal), noise, and delay. We show that a plethora of novel chimera patterns arise if one goes beyond the Kuramoto phase oscillator model. For the FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator with symmetry-breaking coupling various multi-chimera patterns including amplitude chimeras and chimera death occur. To test the robustness of chimera patterns with respect to changes in the structure of the network, regular rings with coupling range R, small-world, and fractal topologies are studied. We also address the robustness of amplitude chimera states in the presence of noise. If delay is added, the lifetime of transient chimeras can be drastically increased.

Amplitude chimeras and chimera death in dynamical networks

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 therefore combines the features of chimera state and oscillation death. Additionally, we show two different transition scenarios from amplitude chimera to chimera death. Moreover, for amplitude chimeras we uncover the mechanism of transition towards in-phase synchronized regime and discuss the role of initial conditions.