Periodic synchronization and chimera in conformist and contrarian oscillators

Emergence of chimeras: An impetus by exceptional points

A nonconservative system with nonreciprocal interaction has been found to reveal exotic features where sudden phase transitions can occur. In this paper, the emanation of a chimera in a network of Stuart-Landau oscillators with nonreciprocal interaction is reported. Note that the spins follow the random discrete distribution. In other words, we pick a random oscillator to rotate clockwise or anticlockwise. At the transition points, we find the spectral singularities in the eigenplane, where eigenvalues coalesce, commonly known as exceptional points. We find that the counterrotational symmetry breaking induced by exceptional points antecedents the occurrence of the chimera. We numerically attest to the findings for two cases of initial conditions, namely, bipartite and random. We also extend our study to a two-dimensional array of nonreciprocally interacting, distributed spins. The findings could have pragmatic implications in the areas of active matter, networks, and photonics.

Discordant synchronization patterns on directed networks of identical phase oscillators with attractive and repulsive couplings

We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling wave and π-states, in addition to the classical fully synchronized and incoherent states. Bistability between all collective states is reported. Analytical results are generally in excellent agreement with simulations; for some parameters and initial conditions, however, we numerically detect chimera-like states which are not captured by the OA theory.

Non-reciprocal phase transitions

Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter^1-6, non-equilibrium systems^7-9, networks of neurons^10,11, social groups with conformist and contrarian members¹², directional interface growth phenomena^13-15 and metamaterials^16-20. Although wave propagation in non-reciprocal media has recently been closely studied^1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points²¹. We describe the emergence of these phases using insights from bifurcation theory^22,23 and non-Hermitian quantum mechanics^24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.

The role of timescale separation in oscillatory ensembles with competitive coupling

We study a heterogeneous population consisting of two groups of oscillatory elements, one with attractive and one with repulsive coupling. Moreover, we set different internal timescales for the oscillators of the two groups and concentrate on the role of this timescale separation in the collective behavior. Our results demonstrate that it may significantly modify synchronization properties of the system, and the implications are fundamentally different depending on the ratio between the group timescales. For the slower attractive group, synchronization properties are similar to the case of equal timescales. However, when the attractive group is faster, these properties significantly change and bistability appears. The other collective regimes such as frozen states and solitary states are also shown to be crucially influenced by timescale separation.

Rhythmic synchronization and hybrid collective states of globally coupled oscillators

Macroscopic rhythms are often signatures of healthy functioning in living organisms, but they are still poorly understood on their microscopic bases. Globally interacting oscillators with heterogeneous couplings are here considered. Thorough theoretical and numerical analyses indicate the presence of multiple phase transitions between different collective states, with regions of bi-stability. Novel coherent phases are unveiled, and evidence is given of the spontaneous emergence of macroscopic rhythms where oscillators’ phases are always found to be self-organized as in Bellerophon states, i.e. in multiple clusters with quantized values of their average frequencies. Due to their rather unconditional appearance, the circumstance is paved that the Bellerophon states grasp the microscopic essentials behind collective rhythms in more general systems of interacting oscillators.

Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control

This paper studies the drive-response synchronization for quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with mixed delays. First, QVSICNN is decomposed into an equivalent real-valued system in order to avoid the non-commutativity of the multiplicity. Then, the existence of almost periodic solutions is obtained based on the Banach fixed point theorem. An novel state-feedback controller is designed to ensure the global exponential almost periodic synchronization. At the end of the paper, an example is given to illustrate the effectiveness of the obtained results.

Local complexity predicts global synchronization of hierarchically networked oscillators

We study the global synchronization of hierarchically-organized Stuart-Landau oscillators, where each subsystem consists of three oscillators with activity-dependent couplings. We considered all possible coupling signs between the three oscillators, and found that they can generate different numbers of phase attractors depending on the network motif. Here, the subsystems are coupled through mean activities of total oscillators. Under weak inter-subsystem couplings, we demonstrate that the synchronization between subsystems is highly correlated with the number of attractors in uncoupled subsystems. Among the network motifs, perfect anti-symmetric ones are unique to generate both single and multiple attractors depending on the activities of oscillators. The flexible local complexity can make global synchronization controllable.

Synchronization and Bellerophon states in conformist and contrarian oscillators

The study of synchronization in generalized Kuramoto models has witnessed an intense boost in the last decade. Several collective states were discovered, such as partially synchronized, chimera, π or traveling wave states. We here consider two populations of globally coupled conformist and contrarian oscillators (with different, randomly distributed frequencies), and explore the effects of a frequency-dependent distribution of the couplings on the collective behaviour of the system. By means of linear stability analysis and mean-field theory, a series of exact solutions is extracted describing the critical points for synchronization, as well as all the emerging stationary coherent states. In particular, a novel non-stationary state, here named as Bellerophon state, is identified which is essentially different from all other coherent states previously reported in the Literature. A robust verification of the rigorous predictions is supported by extensive numerical simulations.

Collective dynamics in two populations of noisy oscillators with asymmetric interactions

We study two intertwined globally coupled networks of noisy Kuramoto phase oscillators that have the same natural frequency but differ in their perception of the mean field and their contribution to it. Such a give-and-take mechanism is given by asymmetric in- and out-coupling strengths which can be both positive and negative. We uncover in this minimal network of networks intriguing patterns of discordance, where the ensemble splits into two clusters separated by a constant phase lag. If it differs from π, then traveling wave solutions emerge. We observe a second route to traveling waves via traditional one-cluster states. Bistability is found between the various collective states. Analytical results and bifurcation diagrams are derived with a reduced system.

Traveling wave in a three-dimensional array of conformist and contrarian oscillators

We consider a system of conformist and contrarian oscillators coupled locally in a three-dimensional cubic lattice and explore collective behavior of the system. The conformist oscillators attractively interact with the neighbor oscillators and therefore tend to be aligned with the neighbors' phase. The contrarian oscillators interact repulsively with the neighbors and therefore tend to be out of phase with them. In this paper, we investigate whether many peculiar dynamics that have been observed in the mean-field system with global coupling can emerge even with local coupling. In particular, we pay attention to the possibility that a traveling wave may arise. We find that the traveling wave occurs due to coupling asymmetry and not by global coupling; this observation confirms that the global coupling is not essential to the occurrence of a traveling wave in the system. The traveling wave can be a mechanism for the coherent rhythm generation of the circadian clock or of hormone secretion in biological systems under local coupling.