Using nonisochronicity to control synchronization in ensembles of nonidentical oscillators

Chimeras in globally coupled oscillators: A review

The surprising phenomenon of chimera in an ensemble of identical oscillators is no more strange behavior of network dynamics and reality. By this time, this symmetry breaking self-organized collective dynamics has been established in many networks, a ring of non-locally coupled oscillators, globally coupled networks, a three-dimensional network, and multi-layer networks. A variety of coupling and dynamical models in addition to the phase oscillators has been used for a successful observation of chimera patterns. Experimental verification has also been done using metronomes, pendula, chemical, and opto-electronic systems. The phenomenon has also been shown to appear in small networks, and hence, it is not size-dependent. We present here a brief review of the origin of chimera patterns restricting our discussions to networks of globally coupled identical oscillators only. The history of chimeras in globally coupled oscillators is older than what has been reported in nonlocally coupled phase oscillators much later. We elaborate the story of the origin of chimeras in globally coupled oscillators in a chronological order, within our limitations, and with brief descriptions of the significant contributions, including our personal experiences. We first introduce chimeras in non-locally coupled and other network configurations, in general, and then discuss about globally coupled networks in more detail.

Strength-dependent perturbation of whole-brain model working in different regimes reveals the role of fluctuations in brain dynamics

Despite decades of research, there is still a lack of understanding of the role and generating mechanisms of the ubiquitous fluctuations and oscillations found in recordings of brain dynamics. Here, we used whole-brain computational models capable of presenting different dynamical regimes to reproduce empirical data's turbulence level. We showed that the model's fluctuations regime fitted to turbulence more faithfully reproduces the empirical functional connectivity compared to oscillatory and noise regimes. By applying global and local strength-dependent perturbations and subsequently measuring the responsiveness of the model, we revealed each regime's computational capacity demonstrating that brain dynamics is shifted towards fluctuations to provide much-needed flexibility. Importantly, fluctuation regime stimulation in a brain region within a given resting state network modulates that network, aligned with previous empirical and computational studies. Furthermore, this framework generates specific, testable empirical predictions for human stimulation studies using strength-dependent rather than constant perturbation. Overall, the whole-brain models fitted to the level of empirical turbulence together with functional connectivity unveil that the fluctuation regime best captures empirical data, and the strength-dependent perturbative framework demonstrates how this regime provides maximal flexibility to the human brain.

Synchronization of three electrochemical oscillators: From local to global coupling

We investigate the formation of synchronization patterns in an oscillatory nickel electrodissolution system in a network obtained by superimposing local and global coupling with three electrodes. We explored the behavior through numerical simulations using kinetic ordinary differential equations, Kuramoto type phase models, and experiments, in which the local to global coupling could be tuned by cross resistances between the three nickel wires. At intermediate coupling strength with predominant global coupling, two of the three oscillators, whose natural frequencies are closer, can synchronize. By adding even a relatively small amount of local coupling (about 9%-25%), a spatially organized partially synchronized state can occur where one of the two synchronized elements is in the center. A formula was derived for predicting the critical coupling strength at which full synchronization will occur independent of the permutation of the natural frequencies of the oscillators over the network. The formula correctly predicts the variation of the critical coupling strength as a function of the global coupling fraction, e.g., with local coupling the critical coupling strength is about twice than that required with global coupling. The results show the importance of the topology of the network on the synchronization properties in a simple three-oscillator setup and could provide guidelines for decrypting coupling topology from identification of synchronization patterns.

Multicluster oscillation death and chimeralike states in globally coupled Josephson Junctions

We observe the multiclustered oscillation death and chimeralike states in an array of Josephson junctions under a combination of self-repulsive and cross-attractive mean-field interaction when each isolated junction is in a bistable state, a coexisting fixed point and an oscillatory state. We locate the parameter landscape of the multiclustered oscillation death and chimeralike states. Alternatively, a purely repulsive mean-field interaction in an array of all oscillatory junctions produces chimeralike states with signatures of metastability in the incoherent subpopulation of junctions.

Chimeralike states in a network of oscillators under attractive and repulsive global coupling

We report chimeralike states in an ensemble of oscillators using a type of global coupling consisting of two components: attractive and repulsive mean-field feedback. We identify the existence of two types of chimeralike states in a bistable Liénard system; in one type, both the coherent and the incoherent populations are in chaotic states (which we refer to as chaos-chaos chimeralike states) and, in another type, the incoherent population is in periodic state while the coherent population has irregular small oscillation. We find a metastable state in a parameter regime of the Liénard system where the coherent and noncoherent states migrate in time from one to another subpopulation. The relative size of the incoherent subpopulation, in the chimeralike states, remains almost stable with increasing size of the network. The generality of the coupling configuration in the origin of the chimeralike states is tested, using a second example of bistable system, the van der Pol-Duffing oscillator where the chimeralike states emerge as weakly chaotic in the coherent subpopulation and chaotic in the incoherent subpopulation. Furthermore, we apply the coupling, in a simplified form, to form a network of the chaotic Rössler system where both the noncoherent and the coherent subpopulations show chaotic dynamics.

Clustering versus non-clustering phase synchronizations

Clustering phase synchronization (CPS) is a common scenario to the global phase synchronization of coupled dynamical systems. In this work, a novel scenario, the non-clustering phase synchronization (NPS), is reported. It is found that coupled systems do not transit to the global synchronization until a certain sufficiently large coupling is attained, and there is no clustering prior to the global synchronization. To reveal the relationship between CPS and NPS, we further analyze the noise effect on coupled phase oscillators and find that the coupled oscillator system can change from CPS to NPS with the increase of noise intensity or system disorder. These findings are expected to shed light on the mechanism of various intriguing self-organized behaviors in coupled systems.

Synchronization of electrochemical oscillators with differential coupling

Experiments are presented to describe the effect of capacitive coupling of two electrochemical oscillators during Ni dissolution in sulfuric acid solution. Equivalent circuit analysis shows that the coupling between the oscillators occurs through the difference between the differentials of the electrode potentials. The differential nature of the coupling introduces strong negative nonisochronicity (i.e., phase shear, strong dependence of the period on the amplitude) in the coupling mechanism with smooth oscillators (under conditions just above a Hopf bifurcation point). Because of the negative nonisochronicity, asymmetrically coupled oscillators exhibit anomalous phase synchronization in the form of frequency difference enhancement. At strong coupling bistability is observed between in-phase and antiphase synchronized states. With relaxation oscillators, in contrast to the resistive coupling where antiphase synchronization can occur, the typical system response with weak coupling is out-of-phase synchronization. When the capacitance is applied on the individual resistors attached to the electrodes the oscillators exhibit weak positive nonisochronicity; this is in contrast with the strong negative nonisochronicity obtained with cross coupling. The proposed coupling configurations reveal the importance of the nonisochronicity level of oscillations for the experimentally observed synchronization patterns and also provide efficient ways of tuning the nonisochronicity level of the oscillations. This latter feature can be exploited to design synchronization features with a combination of resistive (difference) and capacitive (differential) coupling.

Dynamics of electrochemical oscillators with electrode size disparity: asymmetrical coupling and anomalous phase synchronization

Experiments are carried out in dual electrode oscillatory Ni electrodissolution in which the two electrodes have different surface areas. The transition to phase synchronization is analyzed as asymmetrical coupling strength, induced by placing a cross resistance between the electrodes, is varied. It is shown that because of nonisochronicity (phase shear, i.e., strong dependence of period on amplitude) of the oscillators, anomalous phase synchronization effects can be observed: advanced/delayed synchronization and, to a lesser extent, frequency difference enhancement. The type of synchronization is strongly affected by the underlying heterogeneities of the oscillators; in the experiments with a slow driver (large surface area) electrode the synchronization is advanced, with a fast driver electrode the synchronization is delayed with respect to symmetrical coupling. The findings thus reveal that the interplay of asymmetrical coupling with the types of inherent heterogeneities plays an important role for the interpretation of size effects in the dynamical behavior of a nonlinear chemical reaction.

Shear diversity prevents collective synchronization

Large ensembles of heterogeneous oscillators often exhibit collective synchronization as a result of mutual interactions. If the oscillators have distributed natural frequencies and common shear (or nonisochronicity), the transition from incoherence to collective synchronization is known to occur at large enough values of the coupling strength. However, here we demonstrate that shear diversity cannot be counterbalanced by diffusive coupling leading to synchronization. We present the first analytical results for the Kuramoto model with distributed shear and show that the onset of collective synchronization is impossible if the width of the shear distribution exceeds a precise threshold.

Asymmetry-induced effects in coupled phase-oscillator ensembles: Routes to synchronization

A system of two coupled ensembles of phase oscillators can follow different routes to interensemble synchronization. Following a short report of our preliminary results [Phys. Rev. E 78, 025201(R) (2008)], we present a more detailed study of the effects of coupling, noise, and phase asymmetries in coupled phase-oscillator ensembles. We identify five distinct synchronization regions and routes to synchronization that are characteristic of the coupling asymmetry. We show that noise asymmetry induces effects similar to that of coupling asymmetry when the latter is absent. We also find that phase asymmetry controls the probability of occurrence of particular routes to synchronization. Our results suggest that asymmetry plays a crucial role in controlling synchronization within and between oscillator ensembles, and hence that its consideration is vital for modeling real life problems.

Routes to synchrony between asymmetrically interacting oscillator ensembles

We report that asymmetrically interacting ensembles of oscillators follow novel routes to synchrony. These routes seem to be a characteristic feature of coupling asymmetry. We show that they are unaffected by white noise except that the entrainment frequencies are shifted. The probability of occurrence of the routes is determined by phase asymmetry. The identification of these phenomena offers new insight into synchrony between oscillator ensembles and suggest new ways in which it may be controlled.

Experimental evidence of anomalous phase synchronization in two diffusively coupled Chua oscillators

We study the transition to phase synchronization in two diffusively coupled, nonidentical Chua oscillators. In the experiments, depending on the used parameterization, we observe several distinct routes to phase synchronization, including states of either in-phase, out-of-phase, or antiphase synchronization, which may be intersected by an intermediate desynchronization regime with large fluctuations of the frequency difference. Furthermore, we report the first experimental evidence of an anomalous transition to phase synchronization, which is characterized by an initial enlargement of the natural frequency difference with coupling strength. This results in a maximal frequency disorder at intermediate coupling levels, whereas usual phase synchronization via monotonic decrease in frequency difference sets in only for larger coupling values. All experimental results are supported by numerical simulations of two coupled Chua models.

Anomalous phase synchronization in two asymmetrically coupled oscillators in the presence of noise

We study the route to synchronization in two noisy, nonisochronous oscillators. Anomalous phase synchronization arises if both oscillators differ in their respective value of nonisochronicity and it is characterized by a strong detuning of the oscillator frequencies with the onset of coupling. Here we show that anomalous synchronization, both in limit-cycle or chaotic oscillators, can considerably be enlarged under the influence of asymmetrical coupling and noise. In these systems we describe a number of noise induced effects, such as an inversion of the natural frequency difference and coupling induced desynchronization of two identical oscillators. Our results can be explained in terms of a noisy particle in a tilted washboard potential.

Defect-enhanced anomaly in frequency synchronization of asymmetrically coupled spatially extended systems

We analytically establish and numerically show that anomalous frequency synchronization occurs in a pair of asymmetrically coupled chaotic space extended oscillators. The transition to anomalous behaviors is crucially dependent on asymmetries in the coupling configuration, while the presence of phase defects has the effect of enhancing the anomaly in frequency synchronization with respect to the case of merely time chaotic oscillators.

Synchronization of two interacting populations of oscillators

We analyze synchronization between two interacting populations of different phase oscillators. For the important case of asymmetric coupling functions, we find a much richer dynamical behavior compared to that of symmetrically coupled populations of identical oscillators. It includes three types of bistabilities, higher order entrainment, and the existence of states with unusual stability properties. All possible routes to synchronization of the populations are presented and some stability boundaries are obtained analytically. The impact of these findings for neuroscience is discussed.