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We investigate the collective dynamics of a population of XY model-type oscillators, globally coupled via non-separable interactions that are randomly chosen from a positive or negative value, and subject to thermal noise controlled by temperature $T$ . For a finite ratio of positive versus negative coupling, we find that the system at $T = 0$ exhibits a discontinuous, first-order like phase transition from the incoherent to the fully coherent state. We determine the critical threshold for this synchronization transition using a linear stability analysis for the fully coherent state and a heuristic stability argument for the incoherent state. Our theoretical results are supported by extensive numerical simulations which clearly display a first order like transition. Remarkably, the synchronization threshold induced by the type of random coupling considered here is identical to the one found in studies which consider uniform input or output strengths for each oscillator node [Hong and Strogatz, Phys. Rev. E 84, 046202 (2011); H. Hong and S. H. Strogatz, Phys. Rev. Lett. 106, 054102 (2011)]. When thermal noise is present, $T > 0$, the transition from incoherence to the partial coherence is continuous and the critical threshold is now larger compared to the deterministic case, T = 0. We formulate an exact mean-field theory applicable to heterogeneous network structures, based on recent work for the stochastic Kuramoto model using graphon objects representing graphs in the mean-field limit. Applying stability results for the incoherent solution, we derive an exact formula for the synchronization threshold for $T > 0$. In the limit $T \rightarrow 0$ we retrieve the result for the deterministic case with $T = 0$.

We study a simple model of identical "swarmalators," generalizations of phase oscillators that swarm through space. We confine the movements to a one-dimensional (1D) ring and consider distributed (nonidentical) couplings; the combination of these two effects captures an aspect of the more realistic two-dimensional swarmalator model. We discover several collective states which we describe analytically. These states imitate the behavior of vinegar eels, catalytic microswimmers, and other swarmalators which move on quasi-1D rings.

We consider a population of two-dimensional oscillators with random couplings, and explore the collective states. The coupling strength between oscillators is randomly quenched with two values one of which is positive while the other is negative, and the oscillators can spatially {\it{move}} depending on the state variables for phase and position. We find that the system shows the phase transition from the incoherent state to the fully synchronized one at a proper ratio of the number of positive couplings to the total. The threshold is numerically measured, and analytically predicted by the linear stability analysis of the fully synchronized state. It is found that the random couplings induces the long-term state patterns appearing for constant strength. The oscillators move to the places where the randomly quenched couplings work as if annealed. We further observe that the system with mixed randomnesses for quenched couplings shows the combination of the deformed patterns understandable with each annealed averages., 7 pages, 5 figures

Biological floc size is an important reactor microenvironment parameter that is often not experimentally controlled due to a lack of suitable methods. Here, we introduce the Couette-Taylor bioreactor (CTB) as an improved tool for controlling biological floc size, specifically as compared with bubble-column sequencing batch reactors (SBRs). A CTB consists of two concentric walls, either of which may be rotated to induce fluid motion. The induced flow produces hydrodynamic shear which is more uniform than that produced through aeration in SBRs. Because hydrodynamic shear is a major parameter controlling floc size, we hypothesized the ability to better control shear rates within a CTB would enable better-controlled floc sizes. To test this hypothesis, we measured the particle size distributions of activated sludge flocs from CTBs with either inner (iCTB) or outer (oCTB) rotating walls as well as SBRs with varying height to diameter ratios (0.5, 1.1, and 9.4). The rotation speed of the CTBs and aeration rate of the SBRs were varied to produce predicted mean shear rates from 25 to 250 s−1. Further, the shear rate distributions for each experiment were estimated using computational fluid dynamics (CFD). In all SBR experiments, the floc distributions did not significantly vary with shear rate or geometry, likely because shear rates (estimated by CFD) differed much less than originally predicted by theory. In the CTB experiments, the mean particle size decreased proportionally with increased hydrodynamic shear, and iCTBs produced particle size distributions with smaller coefficients of variation than oCTBs (0.3 vs. 0.5–0.7, respectively).

We considered the phase coherence dynamics in a Two-Frequency and Two-Coupling (TFTC) model of coupled oscillators, where coupling strength and natural oscillator frequencies for individual oscillators may assume one of two values (positive/negative). The bimodal distributions for the coupling strengths and frequencies are either correlated or uncorrelated. To study how correlation affects phase coherence, we analyzed the TFTC model by means of numerical simulations and exact dimensional reduction methods allowing to study the collective dynamics in terms of local order parameters [S. Watanabe and S. H. Strogatz, Physica D 74(3-4), 197-253 (1994); E. Ott and T. M. Antonsen, Chaos 18(3), 037113 (2008)]. The competition resulting from distributed coupling strengths and natural frequencies produces nontrivial dynamic states. For correlated disorder in frequencies and coupling strengths, we found that the entire oscillator population splits into two subpopulations, both phase-locked (Lock-Lock) or one phase-locked, and the other drifting (Lock-Drift), where the mean-fields of the subpopulations maintain a constant non-zero phase difference. For uncorrelated disorder, we found that the oscillator population may split into four phase-locked subpopulations, forming phase-locked pairs which are either mutually frequency-locked (Stable Lock-Lock-Lock-Lock) or drifting (Breathing Lock-Lock-Lock-Lock), thus resulting in a periodic motion of the global synchronization level. Finally, we found for both types of disorder that a state of Incoherence exists; however, for correlated coupling strengths and frequencies, incoherence is always unstable, whereas it is only neutrally stable for the uncorrelated case. Numerical simulations performed on the model show good agreement with the analytic predictions. The simplicity of the model promises that real-world systems can be found which display the dynamics induced by correlated/uncorrelated disorder.

Synchronization among arrays of beating cilia is one of the emergent phenomena in biological processes at meso-scopic scales. Strong inter-ciliary couplings modify the natural beating frequencies, $\omega$, of individual cilia to produce a collective motion that moves around a group frequency $\omega_m$. Here we study the thermodynamic cost of synchronizing cilia arrays by mapping their dynamics onto a generic phase oscillator model. The model suggests that upon synchronization the mean heat dissipation rate is decomposed into two contributions, dissipation from each cilium's own natural driving force and dissipation arising from the interaction with other cilia, the latter of which can be interpreted as the one produced by a potential with a time-dependent protocol in the framework of our model. The spontaneous phase-synchronization of beating dynamics of cilia induced by strong inter-ciliary coupling is always accompanied with a significant reduction of dissipation for the cilia population, suggesting that organisms as a whole expend less energy by attaining a temporal order. At the level of individual cilia, however, a population of cilia with $|\omega|< \omega_m$ expend more amount of energy upon synchronization., Comment: 6 pages, 3 figures

Interacting dynamic agents can often exhibit synchronization. It has been reported that the rhythm all agents agree on in the synchronized state could be different from the average of intrinsic rhythms of individual agents. Hinted by such a real-world behavior of the interaction-driven change of the average phase velocity, we propose a modified version of the Kuramoto model, in which the ith oscillator of the phase ϕi interacts with other oscillator j only when the phase difference $${{\phi }}_{{j}}$$ϕj − $${{\phi }}_{i}$$ϕi is in a limited range [−βπ, απ]. From extensive numerical investigations, we conclude that the asymmetric dynamic interaction characterized by β ≠ α leads to the shift of the synchronized frequency with respect to the original distribution of the intrinsic frequency. We also perform and report our computer-based synchronization experiment, which exhibits the expected shift of the synchronized frequency of human participants. In analogy to interacting runners, our result roughly suggests that agents tend to run faster if they are more influenced by runners ahead than behind. We verify the observation by using a simple model of interacting runners.

We study the steady-state patterns of population of the coupled oscillators that sync and swarm, where the interaction distances among oscillators have finite-cutoff in interaction distance. We examine how the static patterns known in the infinite-cutoff are reproduced or deformed, and explore a new static pattern that does not appear until a finite-cutoff is considered. All steady-state patterns of the infinite-cutoff, static sync, static async, and static phase wave are respectively repeated in space for proper finite-cutoff ranges. Their deformation in shape and density takes place for the other finite-cutoff ranges. Bar-like phase wave states are observed, which has not been the case for the infinite-cutoff. All the patterns are investigated via numerical and theoretical analysis., 9 pages, 8 figures

We have frequently encountered the rapid changes that prevalent opinion of the social community is toppled by a new and opposite opinion against the pre-exiting one. To understand this interesting process, mean-field model with infinite-interaction range has been mostly considered in previous studies S. A. Marvel et al., Phys. Rev. Lett. 110, 118702(2012). However, the mean-field interaction range is lack of reality in the sense that any individual cannot interact with all of the others in the community. Based on it, in the present work, we consider a simple model of opinion consensus so-called basic model on the low-dimensional lattices ($d$=1,2) with finite interaction range. The model consists of four types of subpopulations with different opinions: $A, B, AB$, and the zealot of $A$ denoted by $A_c$, following the basic model shown in the work by S. A. Marvel et al.. Comparing with their work, we consider the finite range of the interaction, and particularly reconstruct the lattice structure by adding new links when the two individuals have the distance $

Twisted states with nonzero winding numbers composed of sinusoidally coupled identical oscillators have been observed in a ring. The phase of each oscillator in these states constantly shifts, following its preceding neighbor in a clockwise direction, and the summation of such phase shifts around the ring over $2\ensuremath{\pi}$ characterizes the winding number of each state. In this work, we consider finite-sized $d$-dimensional hypercubic lattices, namely, square ($d=2$) and cubic ($d=3$) lattices with periodic boundary conditions. For identical oscillators, we observe new states in which the oscillators belonging to each line (plane) for $d=2$ ($d=3$) are phase synchronized with nonzero winding numbers along the perpendicular direction. These states can be reduced into twisted states in a ring with the same winding number if we regard each subset of phase-synchronized oscillators as one single oscillator. For nonidentical oscillators with heterogeneous natural frequencies, we observe similar patterns with slightly heterogeneous phases in each line $(d=2)$ and plane $(d=3)$. We show that these states generally appear for random configurations when the global coupling strength is larger than the critical values for the states.

Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact., Collective self-organized behavior can be observed in a variety of systems such as colloids and microswimmers. Here O’Keeffe et al. propose a model of oscillators which move in space and tend to synchronize with neighboring oscillators and outline five types of collective self-organized states.

We consider a mean-field model of coupled phase oscillators with random heterogeneity in the coupling strength. The system that we investigate here is a minimal model that contains randomness in diverse values of the coupling strength, and it is found to return to the original Kuramoto model [Y. Kuramoto, Prog. Theor. Phys. Suppl. 79, 223 (1984)10.1143/PTPS.79.223] when the coupling heterogeneity disappears. According to one recent paper [H. Hong, H. Chaté, L.-H. Tang, and H. Park, Phys. Rev. E 92, 022122 (2015)10.1103/PhysRevE.92.022122], when the natural frequency of the oscillator in the system is "deterministically" chosen, with no randomness in it, the system is found to exhibit the finite-size scaling exponent ν[over ¯]=5/4. Also, the critical exponent for the dynamic fluctuation of the order parameter is found to be given by γ=1/4, which is different from the critical exponents for the Kuramoto model with the natural frequencies randomly chosen. Originally, the unusual finite-size scaling behavior of the Kuramoto model was reported by Hong et al. [H. Hong, H. Chaté, H. Park, and L.-H. Tang, Phys. Rev. Lett. 99, 184101 (2007)10.1103/PhysRevLett.99.184101], where the scaling behavior is found to be characterized by the unusual exponent ν[over ¯]=5/2. On the other hand, if the randomness in the natural frequency is removed, it is found that the finite-size scaling behavior is characterized by a different exponent, ν[over ¯]=5/4 [H. Hong, H. Chaté, L.-H. Tang, and H. Park, Phys. Rev. E 92, 022122 (2015)10.1103/PhysRevE.92.022122]. Those findings brought about our curiosity and led us to explore the effects of the randomness on the finite-size scaling behavior. In this paper, we pay particular attention to investigating the finite-size scaling and dynamic fluctuation when the randomness in the coupling strength is considered.

At how long of an interaction range does an oscillator system behave as a fully-connected one? To answer this question, we consider a system of nonlocally-coupled phase oscillators in one dimension, and explore the effects of a variable interaction range L on collective dynamics. In particular, we investigate the winding-number distribution, paying particular attention to the existence of a twisted wave in the system, and observe that the twisted state vanishes when the interaction range exceeds a critical value. Finite-size scaling of the width of the winding-number distribution reveals that the transition occurs at 2L/N ≈ 0.6, regardless of the system size N. We also show that at the same transition point for the topological twisted state, the phase synchrony in the system becomes partial.

We consider a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths. A fraction $p$ of oscillators are positively coupled, attracting all others, while the remaining fraction $1-p$ are negatively coupled, repelling all others. The frequencies and couplings are deterministically chosen in a manner which correlates them, thereby correlating the two types of disorder in the model. We first explore the effect of this correlation on the system's phase coherence. We find that there is a a critical width $\gamma_c$ in the frequency distribution below which the system spontaneously synchronizes. Moreover, this $\gamma_c$ is independent of $p$. Hence, our model and the traditional Kuramoto model (recovered when $p=1$) have the same critical width $\gamma_c$. We next explore the critical behavior of the system by examining the finite-size scaling and the dynamic fluctuation of the traditional order parameter. We find that the model belongs to the same universality class as the Kuramoto model with deterministically (not randomly) chosen natural frequencies for the case of $p, Comment: 7 pages, 6 figures

We consider a system of coupled swarmalators moving in two dimensional space and explore its collective behavior. Here the swarmalators represent the oscillators that can sync and swarm in space, following the previous study [O’Keeffe et al., Nat. Commun., 8, 1504 (2017)]. The internal state of each swarmalator is represented by its phase angle, and the swarmalators are free to move in the plane according to an equation of motion where the phase and spatial dynamics are coupled with each other. In particular, the phase coupling between the swarmalators is attractive (positive) one, so the coupling makes the swarmalators have their phase difference minimized. The collective behavior of the system is found to be different depending on the extent of the interplay between the phase and spatial dynamics: Specifically, when the extent of the interplay between the phase and spatial dynamics is so weak as to be negligible, the phase dynamics of our system recovers that of the conventional mean-field X Y model. On the other hand, when a certain extent of the interplay is present, the system is found to exhibit the correlated phase where the overall order does not occur. Interestingly, it is found that the correlated phase is the same as the active phase wave found in the system of swarmalators with repulsive phase coupling [O’Keeffe et al., Nat. Commun., 8, 1504 (2017)]. We also find that the system exhibits two different phase transitions: One is the transition from the sync state to the active phase wave state, and the other one is the transition from the active phase wave state to the async state. We perform the finite-size scaling analysis and investigate the transition nature.We consider a system of coupled swarmalators moving in two dimensional space and explore its collective behavior. Here the swarmalators represent the oscillators that can sync and swarm in space, following the previous study [O’Keeffe et al., Nat. Commun., 8, 1504 (2017)]. The internal state of each swarmalator is represented by its phase angle, and the swarmalators are free to move in the plane according to an equation of motion where the phase and spatial dynamics are coupled with each other. In particular, the phase coupling between the swarmalators is attractive (positive) one, so the coupling makes the swarmalators have their phase difference minimized. The collective behavior of the system is found to be different depending on the extent of the interplay between the phase and spatial dynamics: Specifically, when the extent of the interplay between the phase and spatial dynamics is so weak as to be negligible, the phase dynamics of our system recovers that of the conventional mean-field X Y model. ...

We revisit the globally-coupled kinetic Ising model in the presence of an oscillating external magnetic field with an amplitude h0 and a driving angular frequency Ω. In the low-frequency region Ω → 0, the dynamic phase transition is shown to become discontinuous, and the phase boundary is found to be described by h0 ∼ (1 − T ) with the temperature T and a critical temperature of unity in the limit h0 → 0. Our result is complementary to h0 ∼ (1 − T ), previously known for the other limit (1 − T ) Ω. We extend the existing analytic calculation for the continuous transition to the second-order perturbation expansion and find that the dynamic phase transition occurs at lower temperature than the static one, which was not captured by the existing first-order calculation. The stochastic resonance behavior is also discussed in relation to the nature of the dynamic phase transition.

We consider a mean-field model of coupled phase oscillators with quenched disorder in the coupling strengths and natural frequencies. When these two kinds of disorder are uncorrelated (and when the positive and negative couplings are equal in number and strength), it is known that phase coherence cannot occur and synchronization is absent. Here we explore the effects of correlating the disorder. Specifically, we assume that any given oscillator either attracts or repels all the others, and that the sign of the interaction is deterministically correlated with the given oscillator's natural frequency. For symmetrically correlated disorder with zero mean, we find that the system spontaneously synchronizes, once the width of the frequency distribution falls below a critical value. For asymmetrically correlated disorder, the model displays coherent traveling waves: the complex order parameter becomes nonzero and rotates with constant frequency different from the system's mean natural frequency. Thus, in both cases, correlated disorder can trigger phase coherence., 6 pages, 6 figures

We numerically study the critical behavior of the one-dimensional XY model of the size N with variable interaction range L. As expected, the standard local order parameter of the magnetization is shown to well detect the mean-field type transition which occurs at any nonzero value of L/N. The system is particularly interesting since the underlying one-dimensional structure allows us to study the topological excitation of the winding number across the whole system even though the system shares the mean-field transition with the globally-coupled system. We propose a novel nonlocal order parameter based on the width of the winding number distribution which exhibits a clear signature of the transition nature of the system., Comment: 6 pages, 6 figures

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.

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population of size $N$, we study two ways of sampling the intrinsic frequencies according to the {\it same} given unimodal distribution $g(\omega)$. In the `{\em random}' case, frequencies are generated independently in accordance with $g(\omega)$, which gives rise to oscillator number fluctuation within any given frequency interval. In the `{\em regular}' case, the $N$ frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasi-uniformly spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case, but is violated in the random case. In this last case, a self-consistent mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order parameter but not its critical amplitude.

We study the synchronization phenomena in systems of globally coupled oscillators, each possessing finite inertia, with particular attention to the noise effects. The self-consistency equation for the order parameter as well as the probability distribution is obtained from the Smoluchowski equation, and analyzed in the presence of thermal noise. It is found that the hysteresis present in the system without noise disappears as the thermal noise comes into the system. Numerical simulations are also performed to give results generally consistent with the analytical ones.

We study analytically the synchronization phenomena in a set of globally coupled oscillators under external periodic driving, with emphasis on the effects of small inertia. We examine in detail both the integer and fractional mode locking present in the system with inertia, and derive the self-consistency equation for the order parameter, which reveals variation of the magnitude of the order parameter according to the external periodic driving. In particular, it is found that the inertia induces discontinuous transitions between the coherent and incoherent states.

We consider a system of phase oscillators with random intrinsic frequencies coupled through sparse random networks, and investigate how the connectivity disorder affects the nature of collective synchronization transitions. Various distribution types of intrinsic frequencies are considered: uniform, unimodal, and bimodal distribution. We employ a heterogeneous mean-field approximation based on the annealed networks and also perform numerical simulations on the quenched Erdos-Renyi networks. We find that the connectivity disorder drastically changes the nature of the synchronization transitions. In particular, the quenched randomness completely wipes away the diversity of the transition nature and only a continuous transition appears with the same mean-field exponent for all types of frequency distributions. The physical origin of this unexpected result is discussed., Comment: 9 pages

We consider a system of phase oscillators that couple with both attractive and repulsive interaction under a pinning force and explore collective behavior of the system. The oscillators can be divided into two subpopulations of "conformist" oscillators with attractive interaction and "contrarian" ones with repulsive interaction. We find that the interplay between the pinning force and the opposite relationship of the conformist and contrarian oscillators induce peculiar dynamic states: periodic synchronization, breathing chimera, and fully pinned state depending on the fraction of the conformists. Using the Watanabe-Strogatz transformation, we reduce the dynamics into a low-dimensional one and find that the above dynamic states are generated from the reduced dynamics.

Pancreatic islets, controlling glucose homeostasis, consist of \alpha, \beta, and \delta\ cells. It has been observed that \alpha\ and \beta\ cells generate out-of-phase synchronization in the release of glucagon and insulin, counter-regulatory hormones for increasing and decreasing glucose levels, while \beta\ and \delta\ cells produce in-phase synchronization in the release of the insulin and somatostatin. Pieces of interactions between the islet cells have been observed for a long time, although their physiological role as a whole has not been explored yet. We model the synchronized hormone pulses of islets with coupled phase oscillators that incorporate the observed cellular interactions. The integrated model shows that the interaction from \beta\ to \delta\ cells, of which sign has controversial reports, should be positive to reproduce the in-phase synchronization between \beta\ and \delta\ cells. The model also suggests that \delta\ cells help the islet system flexibly respond to changes of glucose environment., Comment: 6 pages, 3 figures, accepted in PRE

We consider a costly bilingualism model in which one can take two strategies in parallel. We investigate how a single zealot triggers the cascading behavior and how the compatibility of the two strategies affects when interacting patterns change. First, the role of the interaction range on the cascading is studied by increasing the range from local to global. We find that people sometimes do not favor to take the superior strategy even though its payoff is higher than that of the inferior one. This is found to be caused by the local interactions rather than the global ones. Applying this model to social networks, we find that the location of the zealot is also important for larger cascading in heterogeneous networks., 5 pages, 3 figures, 2 tables

We consider one typical system of oscillators coupled through disordered link configurations in networks, i.e., a finite population of coupled phase oscillators with distributed intrinsic frequencies on a random network. We investigate collective synchronization behavior, paying particular attention to link-disorder fluctuation effects on the synchronization transition and its finite-size scaling (FSS). Extensive numerical simulations as well as the mean-field analysis have been performed. We find that link-disorder fluctuations effectively induce {\em uncorrelated random} fluctuations in frequency, resulting in the FSS exponent $\bar\nu=5/2$, which is identical to that in the globally coupled case (no link disorder) with frequency-disorder fluctuations., Comment: 5 pages, 5 figures

Some of the most pivotal moments in intellectual history occur when a new ideology sweeps through a society, supplanting an established system of beliefs in a rapid revolution of thought. Yet in many cases the new ideology is as extreme as the old. Why is it then that moderate positions so rarely prevail? Here, in the context of a simple model of opinion spreading, we test seven plausible strategies for deradicalizing a society and find that only one of them significantly expands the moderate subpopulation without risking its extinction in the process., Comment: 9 pages, 5 figures

We consider a variant of the Kuramoto model of coupled oscillators in which both attractive and repulsive pairwise interactions are allowed. The sign of the coupling is assumed to be a characteristic of a given oscillator. Specifically, some oscillators repel all the others, thus favoring an antiphase relationship with them. Other oscillators attract all the others, thus favoring an in-phase relationship. The Ott-Antonsen ansatz is used to derive the exact low-dimensional dynamics governing the system's long-term macroscopic behavior. The resulting analytical predictions agree with simulations of the full system. We explore the effects of changing various parameters, such as the width of the distribution of natural frequencies and the relative strengths and proportions of the positive and negative interactions. For the particular model studied here we find, unexpectedly, that the mixed interactions produce no new effects. The system exhibits conventional mean-field behavior and displays a second-order phase transition like that found in the original Kuramoto model. In contrast to our recent study of a different model with mixed interactions [Phys. Rev. Lett. 106, 054102 (2011)], the $\ensuremath{\pi}$ state and traveling-wave state do not appear for the coupling type considered here.

We demonstrate that a large ensemble of noiseless globally coupled-pinned oscillators is capable of rectifying spatial disorder with spontaneous current activated through a dynamical phase transition mechanism, either of first or second order, depending on the profile of the pinning potential. In the presence of an external weak drive, the same collective mechanism can result in an absolute negative mobility, which, though not immediately related to symmetry breaking, is most prominent at the phase transition. Our results apply to a tug-of-war by competing molecular motors for bidirectional cargo transport.

We consider a variant of the Kuramoto model, in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These "contrarian" oscillators tend to align in antiphase with the mean field, whereas the positively coupled "conformist" oscillators favor an in-phase relationship. The interplay between these effects can lead to rich dynamics. In addition to a splitting of the population into two diametrically opposed factions, the system can also display traveling waves, complete incoherence, and a blurred version of the two-faction state. Exact solutions for these states and their bifurcations are obtained by means of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously, this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies., Comment: 7 pages, 4 figures, to appear in Phys. Rev. E

We consider a generalization of the Kuramoto model in which the oscillators are coupled to the mean field with random signs. Oscillators with positive coupling are ``conformists''; they are attracted to the mean field and tend to synchronize with it. Oscillators with negative coupling are ``contrarians''; they are repelled by the mean field and prefer a phase diametrically opposed to it. The model is simple and exactly solvable, yet some of its behavior is surprising. Along with the stationary states one might have expected (a desynchronized state, and a partially-synchronized state, with conformists and contrarians locked in antiphase), it also displays a traveling wave, in which the mean field oscillates at a frequency different from the population's mean natural frequency.

We consider globally coupled random frequency oscillators under thermal noise, and explore the synchronization transition with its critical behavior near the transition. In particular, we focus on the finite-size scaling behavior of the synchronization, and investigate how the thermal noise affects the correlation size exponent ν of the synchronized oscillators. Extensive numerical simulations as well as mean-field analysis have been performed. We find that the correlation size exponent changes from ν=5/2 without thermal noise to ν=2 with strong thermal noise, where the value ν=2 is the same as that for the usual equilibrium systems described by the Ginzburg-Landau mean-field theory. In order to see the effects of thermal fluctuation further, we remove the frequency-disorder fluctuations originating from the different realizations of natural frequencies of the oscillators, and examine the finite-size scaling behavior for the case only with the thermal fluctuation. It is found that ν becomes 2 at much weak thermal noise strength, which implies that even very weak thermal fluctuations may lead to ν=2 when frequency-disorder fluctuations are absent.

We study the nonequilibrium phenomena of a coupled active rotator model in complex networks. From a numerical Langevin simulation, we find the peculiar phase transition not only on globally connected network but also on other complex networks and reveal the corresponding phase diagram. In this model, two phases — stationary and quasi-periodic moving phases — are observed, in which microscopic dynamics are thoroughly investigated. We extend our study to the non-identical oscillators and the more heterogeneous degree distribution of complex networks.

We investigate how we can improve the synchronizability of complex networks simply by changing the link direction while conserving the local link weights and topology. Performing the linear stability analysis of synchronization and numerical simulation of the Kuramoto model in the directed networks, we find that while a random assignment of link directions generally weakens the degree of synchronization, a properly organized directionality can systematically enhance the network synchronization. In this respect, we suggest a simple method of changing the link direction according to the larger residual degree starting from small residual degree nodes. This result provides plausible applications to control the synchronizability of systems in various fields.

We study collective synchronization in a large number of coupled oscillators on various complex networks. In particular, we focus on the relaxation dynamics of the synchronization, which is important from the viewpoint of information transfer or the dynamics of system recovery from a perturbation. We measure the relaxation time tau that is required to establish global synchronization by varying the structural properties of the networks. It is found that the relaxation time in a strong-coupling regime (KKc) logarithmically increases with network size N , which is attributed to the initial random phase fluctuation given by O(N-1/2) . After elimination of the initial-phase fluctuation, the relaxation time is found to be independent of the system size; this implies that the local interaction that depends on the structural connectivity is irrelevant in the relaxation dynamics of the synchronization in the strong-coupling regime. The relaxation dynamics is analytically derived in a form independent of the system size, and it exhibits good consistency with numerical simulations. As an application, we also explore the recovery dynamics of the oscillators when perturbations enter the system.

The onset of synchronization in a system of random frequency oscillators coupled through a random network is investigated. Using a mean-field approximation, we characterize sample-to-sample fluctuations for networks of finite size, and derive the corresponding scaling properties in the critical region. For scale-free networks with the degree distribution $P(k)\sim k^{-\gamma}$ at large $k$, we found that the finite size exponent $\bar{\nu}$ takes on the value 5/2 when $\gamma>5$, the same as in the globally coupled Kuramoto model. For highly heterogeneous networks ($3, Comment: 7 pages

A finite-size-scaling (FSS) theory is proposed for various models in complex networks. In particular, we focus on the FSS exponent, which plays a crucial role in analyzing numerical data for finite-size systems. Based on the droplet-excitation (hyperscaling) argument, we conjecture the values of the FSS exponents for the Ising model, the susceptible-infected-susceptible model, and the contact process, all of which are confirmed reasonably well in numerical simulations.

The entrainment transition of coupled random frequency oscillators is revisited. The Kuramoto model (global coupling) is shown to exhibit unusual sample-dependent finite size effects leading to a correlation size exponent $\bar��=5/2$. Simulations of locally coupled oscillators in $d$-dimensions reveal two types of frequency entrainment: mean-field behavior at $d>4$, and aggregation of compact synchronized domains in three and four dimensions. In the latter case, scaling arguments yield a correlation length exponent $��=2/(d-2)$, in good agreement with numerical results., published version

Recently, Castellano and Pastor-Satorras [1] utilized the finite size scaling (FSS) theory to analyze simulation data for the contact process (CP) on scale-free networks (SFNs) and claimed that its absorbing critical behavior is not consistent with the mean-field (MF) prediction. Furthermore, they pointed out large density fluctuations at highly connected vertices as a possible origin for non-MF critical behavior. In this Comment, we propose a scaling theory for relative density fluctuations in the spirit of the MF theory, which turns out to explain simulation data perfectly well. We also measure the value of the critical density decay exponent, which agrees well with the MF prediction. Our results strongly support that the CP on SFNs still exhibits a MF-type critical behavior., Comment: 1 page, 2 figures, typos are corrected

We investigate the effects of quenched disorder on the coherent response in a driven system of coupled oscillators. In particular, the interplay between quenched noise and periodic driving is explored, with particular attention to the possibility of resonance. The phase velocity is examined as the response of the system; revealed is the enhancement of the fraction of oscillators locked to the periodic driving, displaying resonance behavior. It is thus concluded that resonance behavior may also be induced by quenched disorder which does not have time-dependent correlations., 4 pages, 3 figures, to appear in Phys. Rev. E

We study collective behavior of locally coupled limit-cycle oscillators with random intrinsic frequencies, spatially extended over $d$-dimensional hypercubic lattices. Phase synchronization as well as frequency entrainment are explored analytically in the linear (strong-coupling) regime and numerically in the nonlinear (weak-coupling) regime. Our analysis shows that the oscillator phases are always desynchronized up to $d=4$, which implies the lower critical dimension $d_{l}^{P}=4$ for phase synchronization. On the other hand, the oscillators behave collectively in frequency (phase velocity) even in three dimensions ($d=3$), indicating that the lower critical dimension for frequency entrainment is $d_{l}^{F}=2$. Nonlinear effects due to periodic nature of limit-cycle oscillators are found to become significant in the weak-coupling regime: So-called {\em runaway oscillators} destroy the synchronized (ordered) phase and there emerges a fully random (disordered) phase. Critical behavior near the synchronization transition into the fully random phase is unveiled via numerical investigation. Collective behavior of globally-coupled oscillators is also examined and compared with that of locally coupled oscillators., Comment: 18 pages, 18 figures