Synchronization of oscillators with long range interaction: Phase transition and anomalous finite size effects

Fate of vortex-synchronized state in oscillator networks with node defects

We investigate synchronization behaviors of a Kuramoto oscillator network with a two-dimensional square-lattice configuration. We show that the oscillator network can reach a phase-locking vortex synchronized state in the long time limit starting from random initial oscillator phases sampled according to the von Mises distribution characterized by a zero mean and a finite concentration parameter. We further reveal that the stability of the vortex synchronized state is sensitive to the presence of local node defects, in contrast to the usual knowledge that oscillator networks should exhibit robustness against local perturbations. Moreover, we explore the behaviors of the vortex synchronized state in networks with an additional temporal white noise on the oscillator phases or a spatial noise due to randomly distributed oscillator frequencies. Interestingly, we find that the vortex synchronized state can become immune to local node defects when the variance of spatial noise is above a certain threshold, suggesting a beneficial role of usually unwanted spatial noise in protecting vortex-synchronized networks.

Dynamical robustness of complex networks subject to long-range connectivity

In spite of a few attempts in understanding the dynamical robustness of complex networks, this extremely important subject of research is still in its dawn as compared to the other dynamical processes on networks. We hereby consider the concept of long-range interactions among the dynamical units of complex networks and demonstrate for the first time that such a characteristic can have noteworthy impacts on the dynamical robustness of networked systems, regardless of the underlying network topology. We present a comprehensive analysis of this phenomenon on top of diverse network architectures. Such dynamical damages being able to substantially affect the network performance, determining mechanisms that boost the robustness of networks becomes a fundamental question. In this work, we put forward a prescription based upon self-feedback that can efficiently resurrect global rhythmicity of complex networks composed of active and inactive dynamical units, and thus can enhance the network robustness. We have been able to delineate the whole proposition analytically while dealing with all d -path adjacency matrices, having an excellent agreement with the numerical results. For the numerical computations, we examine scale-free networks, Watts–Strogatz small-world model and also Erdös–Rényi random network, along with Landau–Stuart oscillators for casting the local dynamics.

Neuronal synchronization in long-range time-varying networks

We study synchronization in neuronal ensembles subject to long-range electrical gap junctions which are time-varying. As a representative example, we consider Hindmarsh-Rose neurons interacting based upon temporal long-range connections through electrical couplings. In particular, we adopt the connections associated with the direct 1-path network to form a small-world network and follow-up with the corresponding long-range network. Further, the underlying direct small-world network is allowed to temporally change; hence, all long-range connections are also temporal, which makes the model much more realistic from the neurological perspective. This time-varying long-range network is formed by rewiring each link of the underlying 1-path network stochastically with a characteristic rewiring probability p_r, and accordingly all indirect k(>1)-path networks become temporal. The critical interaction strength to reach complete neuronal synchrony is much lower when we take up rapidly switching long-range interactions. We employ the master stability function formalism in order to characterize the local stability of the state of synchronization. The analytically derived stability condition for the complete synchrony state agrees well with the numerical results. Our work strengthens the understanding of time-varying long-range interactions in neuronal ensembles.

Phase synchronization in the two-dimensional Kuramoto model: Vortices and duality

We study a system of Kuramoto oscillators arranged on a two-dimensional periodic lattice where the oscillators interact with their nearest neighbors, and all oscillators have the same natural frequency. The initial phases of the oscillators are chosen to be distributed uniformly between (-π,π]. During the relaxation process to the final stationary phase, we observe different features in the phase field of the oscillators: initially, the state is randomly oriented, then clusters form. As time evolves, the size of the clusters increases and vortices that constitute topological defects in the phase field form in the system. These defects, being topological, annihilate in pairs; i.e., a given defect annihilates if it encounters another defect with opposite polarity. Finally, the system ends up either in a completely phase synchronized state in case of complete annihilation or a metastable phase locked state characterized by presence of vortices and antivortices. The basin volumes of the two scenarios are estimated. Finally, we carry out a duality transformation similar to that carried out for the XY model of planar spins on the Hamiltonian version of the Kuramoto model to expose the underlying vortex structure.

Chimera patterns induced by distance-dependent power-law coupling in ecological networks

This paper reports the occurrence of several chimera patterns and the associated transitions among them in a network of coupled oscillators, which are connected by a long-range interaction that obeys a distance-dependent power law. This type of interaction is common in physics and biology and constitutes a general form of coupling scheme, where by tuning the power-law exponent of the long-range interaction the coupling topology can be varied from local via nonlocal to global coupling. To explore the effect of the power-law coupling on collective dynamics, we consider a network consisting of a realistic ecological model of oscillating populations, namely the Rosenzweig-MacArthur model, and show that the variation of the power-law exponent mediates transitions between spatial synchrony and various chimera patterns. We map the possible spatiotemporal states and their scenarios that arise due to the interplay between the coupling strength and the power-law exponent.

Synchronization and plateau splitting of coupled oscillators with long-range power-law interactions

We investigate synchronization and plateau splitting of coupled oscillators on a one-dimensional lattice with long-range interactions that decay over distance as a power law. We show that in the thermodynamic limit the dynamics of systems of coupled oscillators with power-law exponent α≤1 is identical to that of the all-to-all coupling case. For α>1, oscillatory behavior of the phase coherence appears as a result of single plateau splitting into multiple plateaus. A coarse-graining method is used to investigate the onset of plateau splitting. We analyze a simple oscillatory state formed by two plateaus in detail and propose a systematic approach to predict the onset of plateau splitting. The prediction of breaking points of plateau splitting is in quantitatively good agreement with numerical simulations.

One-dimensional lattice of oscillators coupled through power-law interactions: continuum limit and dynamics of spatial Fourier modes

We study synchronization in a system of phase-only oscillators residing on the sites of a one-dimensional periodic lattice. The oscillators interact with a strength that decays as a power law of the separation along the lattice length and is normalized by a size-dependent constant. The exponent α of the power law is taken in the range 0≤α<1. The oscillator frequency distribution is symmetric about its mean (taken to be zero) and is nonincreasing on [0,∞). In the continuum limit, the local density of oscillators evolves in time following the continuity equation that expresses the conservation of the number of oscillators of each frequency under the dynamics. This equation admits as a stationary solution the unsynchronized state uniform both in phase and over the space of the lattice. We perform a linear stability analysis of this state to show that when it is unstable, different spatial Fourier modes of fluctuations have different stability thresholds beyond which they grow exponentially in time with rates that depend on the Fourier modes. However, numerical simulations show that at long times all the nonzero Fourier modes decay in time, while only the zero Fourier mode (i.e., the "mean-field" mode) grows in time, thereby dominating the instability process and driving the system to a synchronized state. Our theoretical analysis is supported by extensive numerical simulations.

Many-body theory of synchronization by long-range interactions

Synchronization of coupled oscillators on a d-dimensional lattice with the power-law coupling G(r) = g0/rα and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of ϵ = α/d-1. For α ≤ d, the system exhibits a sharp synchronization transition as described by the conventional mean-field theory. For α > d, the transition is smeared by the quenched disorder, and the macroscopic order parameter ψ decays slowly with g0 as |ψ| ∝ g(0)(2).

Vortices and the entrainment transition in the two-dimensional Kuramoto model

We study synchronization in the two-dimensional lattice of coupled phase oscillators with random intrinsic frequencies. When the coupling K is larger than a threshold K{E} , there is a macroscopic cluster of frequency-synchronized oscillators. We explain why the macroscopic cluster disappears at K{E} . We view the system in terms of vortices, since cluster boundaries are delineated by the motion of these topological defects. In the entrained phase (K>K{E}) , vortices move in fixed paths around clusters, while in the unentrained phase (K<K{E}) , vortices sometimes wander off. These deviant vortices are responsible for the disappearance of the macroscopic cluster. The regularity of vortex motion is determined by whether clusters behave as single effective oscillators. The unentrained phase is also characterized by time-dependent cluster structure and the presence of chaos. Thus, the entrainment transition is actually an order-chaos transition. We present an analytical argument for the scaling K{E}∼K{L} for small lattices, where K{L} is the threshold for phase locking. By also deriving the scaling K{L}∼log N , we thus show that K{E}∼log N for small N , in agreement with numerics. In addition, we show how to use the linearized model to predict where vortices are generated.

Synchronization of oscillators with long-range power law interactions

We present analytical calculations and numerical simulations for the synchronization of oscillators interacting via a long-range power law interaction on a one-dimensional lattice. We have identified the critical value of the power law exponent α(c) across which a transition from a synchronized to an unsynchronized state takes place for a sufficiently strong but finite coupling strength in the large system limit. We find α(c)=3/2. Frequency entrainment and phase ordering are discussed as a function of α≥1 . The calculations are performed using an expansion about the aligned phase state (spin-wave approximation) and a coarse graining approach. We also generalize the spin-wave results to the d -dimensional problem.

Synchrony and stability of food webs in metacommunities

Synchrony has fundamental but conflicting implications for the persistence and stability of food webs at local and regional scales. In a constant environment, compensatory dynamics between species can maintain food web stability, but factors that synchronize population fluctuations within and between communities are expected to be destabilizing. We studied the dynamics of a food web in a metacommunity to determine how environmental variability and dispersal affect stability by altering compensatory dynamics and average species abundance. When dispersal rate is high, weak correlated environmental fluctuations promote food web stability by reducing the amplitude of compensatory dynamics. However, when dispersal rate is low, weak environmental fluctuations reduce food web stability by inducing intraspecific synchrony across communities. Irrespective of dispersal rate, strong environmental fluctuations disrupt compensatory dynamics and decrease stability by inducing intermittent correlated fluctuations between consumers in local food webs, which reduce both total consumer abundance and predator abundance. Strong correlated environmental fluctuations lead to (i) spatially asynchronous and highly correlated local consumer dynamics when dispersal is low and (ii) spatially synchronous but intermediate local consumer correlation when dispersal is high. By controlling intraspecific synchrony, dispersal mediates the capacity of strong environmental fluctuations to disrupt compensatory dynamics at both local and metacommunity scales.

Resonances and entrainment breakup in Kuramoto models with multimodal frequency densities

We characterize some intriguing aspects of the entrainment behavior of coupled oscillators by means of a perturbation analysis of the partially synchronized solution of the classical Kuramoto-Sakaguchi model. The analysis reveals that partial entrainment may disappear with increasing coupling strength. It also predicts the occurrence of resonances: partial entrainment is induced in oscillators with natural frequencies in specific intervals not corresponding to high oscillator densities. The results are illustrated by simulations.

Reentrant synchronization and pattern formation in pacemaker-entrained Kuramoto oscillators

We study phase entrainment of Kuramoto oscillators under different conditions on the interaction range and the natural frequencies. In the first part the oscillators are entrained by a pacemaker acting like an impurity or a defect. We analytically derive the entrainment frequency for arbitrary interaction range and the entrainment threshold for all-to-all couplings. For intermediate couplings our numerical results show a reentrance of the synchronization transition as a function of the coupling range. The origin of this reentrance can be traced back to the normalization of the coupling strength. In the second part we consider a system of oscillators with an initial gradient in their natural frequencies, extended over a one-dimensional chain or a two-dimensional lattice. Here it is the oscillator with the highest natural frequency that becomes the pacemaker of the ensemble, sending out circular waves in oscillator-phase space. No asymmetric coupling between the oscillators is needed for this dynamical induction of the pacemaker property nor need it be distinguished by a gap in the natural frequency.

Synchronization in large directed networks of coupled phase oscillators

We study the emergence of collective synchronization in large directed networks of heterogeneous oscillators by generalizing the classical Kuramoto model of globally coupled phase oscillators to more realistic networks. We extend recent theoretical approximations describing the transition to synchronization in large undirected networks of coupled phase oscillators to the case of directed networks. We also consider the case of networks with mixed positive-negative coupling strengths. We compare our theory with numerical simulations and find good agreement.

Regularization of dynamics in an ensemble of nondiffusively coupled chaotic elements

We investigate the dynamics in an ensemble of chaotic elements with nondiffusive coupling. First, we analyze the case of global coupling. The type of coupling we consider leads to the suppression of oscillations in the whole ensemble at a high coupling strength. A distinct feature of this transition from high-dimensional chaos at a low coupling strength to the stationary state is that there is no partially ordered phase characterized by a large number of coexisting synchronized clusters. A two-cluster mode emerges abruptly, replacing the asynchronous mode. We focus on the influence of connectivity on the dynamics in the two-cluster modes and their domains of existence. We introduce a parameter that characterizes the connectivity: the range of coupling. Our computational and analytical results indicate that the most significant changes in the dynamics occur in the case of local coupling, when extra connections are added. By contrast, if the range of coupling is high, even substantial changes in this range have a small influence on the dynamics.

Repulsive synchronization in an array of phase oscillators

We study the dynamics of a repulsively coupled array of phase oscillators. For an array of globally coupled identical oscillators, repulsive coupling results in a family of synchronized regimes characterized by zero mean field. If the number of oscillators is sufficiently large, phase locking among oscillators is destroyed, independently of the coupling strength, when the oscillators' natural frequencies are not the same. In locally coupled networks, however, phase locking occurs even for nonidentical oscillators when the coupling strength is sufficiently strong.

Onset of synchronization in large networks of coupled oscillators

We study the transition from incoherence to coherence in large networks of coupled phase oscillators. We present various approximations that describe the behavior of an appropriately defined order parameter past the transition and generalize recent results for the critical coupling strength. We find that, under appropriate conditions, the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix. We show how, with an additional assumption, a mean-field approximation recently proposed is recovered from our results. We test our theory with numerical simulations and find that it describes the transition when our assumptions are satisfied. We find that our theory describes the transition well in situations in which the mean-field approximation fails. We study the finite-size effects caused by nodes with small degree and find that they cause the critical coupling strength to increase.

Chromatid transport by pantographic motors (PMS)

The paper discusses a system of hinged leverages functioning as (vectorial) pantographic motors (PMs). Mediated by ratchets, PMs are suited for ferrying mammalian chromatids to mitotic antipoles. Several self-rectifying modes allow PMs to accurately partition genomes with minimal risk of engendering aneuploidy.

Analytical results for coupled-map lattices with long-range interactions

We obtain exact analytical results for lattices of maps with couplings that decay with distance as r(-alpha). We analyze the effect of the coupling range on the system dynamics through the Lyapunov spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov spectrum for a completely synchronized state is analytically obtained. The critical line characterizing the synchronization transition is determined from the expression for the largest transversal Lyapunov exponent. In particular, it is shown that in the thermodynamical limit, such transition is only possible for sufficiently long-range interactions, namely, for alpha<alpha(c)=d, where d is the lattice dimension.