It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.
A general stability analysis is presented for the determination of the transition from incoherent to coherent behavior in an ensemble of globally coupled, heterogeneous, continuous-time dynamical systems. The formalism allows for the simultaneous presence of ensemble members exhibiting chaotic and periodic behavior, and, in a special case, yields the Kuramoto model for globally coupled periodic oscillators described by a phase. Numerical experiments using different types of ensembles of Lorenz equations with a distribution of parameters are presented.
We demonstrate emergence of a complex state in a homogeneous ensemble of globally coupled identical oscillators, reminiscent of chimera states in locally coupled oscillator lattices. In this regime some part of the ensemble forms a regularly evolving cluster, while all other units irregularly oscillate and remain asynchronous. We argue that chimera emerges because of effective bistability which dynamically appears in the originally monostable system due to internal delayed feedback in individual units. Additionally, we present two examples of chimeras in bistable systems with frequency-depende…
We consider an infinite network of globally-coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior i…
Experiments are carried out with a globally coupled, externally forced population of limit-cycle electrochemical oscillators with an approximately unimodal distribution of heterogeneities. Global coupling induces mutually entrained (at frequency $ω_{1}$) states; periodic forcing produces forced-entrained ($ω_{\mathrm{F}}$) states. As a result of the interaction of mutual and forced entrainment, resonant cluster states occur with equal spacing of frequencies that have discretized frequencies following a resonance rule $ω_{n}\cong nω_{1}-(n-1)ω_{\mathrm{F}}$. Resonance clustering requires an opt…
Large systems of coupled oscillators subjected to a periodic external drive occur in many situations in physics and biology. Here the simple, paradigmatic case of equal-strength, all-to-all sine-coupling of phase oscillators subject to a sinusoidal external drive is considered. The stationary states and their stability are determined. Using the stability information and numerical experiments, parameter space phase diagrams showing when different types of system behavior apply are constructed, and the bifurcations marking transitions between different types of behavior are delineated. The analy…
In globally coupled ensembles of identical oscillators so-called chimera states can be observed. The chimera state is a symmetry-broken regime, where a subset of oscillators forms a cluster, a synchronized population, while the rest of the system remains a collection of non-synchronized, scattered units. We describe here a blinking chimera regime in an ensemble of seven globally coupled rotators (Kuramoto oscillators with inertia). It is characterized by a death-birth process, where a long-term stable cluster of four oscillators suddenly dissolves and is very quickly reborn with a new, reshuff…
From rhythmic physiological processes to the collective behaviors of technological and natural networks, coherent phases of interacting oscillators are the foundation of the events' coordination leading a system to behave cooperatively. We unveil the existence of a new of such states, occurring in globally coupled nonidentical oscillators in the proximity of the point where the transition from the system's incoherent to coherent phase converts from explosive to continuous. In such a state, oscillators form quantized clusters, where they are neither phase- nor frequency-locked. Oscillators' ins…
Many studies of synchronization properties of coupled oscillators, based on the classical Kuramoto approach, focus on ensembles coupled via a mean field. Here we introduce a setup of Kuramoto-type phase oscillators coupled via two mean fields. We derive stability properties of the incoherent state and find traveling wave solutions with different locking patterns; stability properties of these waves are found numerically. Mostly nontrivial states appear when the two fields compete, i.e. one tends to synchronize oscillators while the other one desynchronizes them. Here we identify normal branche…
We argue that a discrete Shilnikov attractor exists in the system of five identical globally coupled phase oscillators with biharmonic coupling. We explain the scenario that leads to birth of this kind of attractor and numerically illustrate the sequence of bifurcations that supports our statement.
Cooperative effects of periodic force and noise in globally Cooperative effects of periodic force and noise in globally coupled systems are studied using a nonlinear diffusion equation for the number density. The amplitude of the order parameter oscillation is enhanced in an intermediate range of noise strength for a globally coupled bistable system, and the order parameter oscillation is entrained to the external periodic force in an intermediate range of noise strength. These enhancement phenomena of the response of the order parameter in the deterministic equations are interpreted as stocha…
Oscillatory systems with long-range or global coupling offer promising insight into the interplay between high-dimensional (or microscopic) chaotic motion and collective interaction patterns. Within this paper, we use Lyapunov analysis to investigate whether chimera states in globally coupled Stuart-Landau (SL) oscillators exhibit collective degrees of freedom. We compare two types of chimera states, which emerge in SL ensembles with linear and nonlinear global coupling, respectively, the latter introducing a constraint that conserves the oscillation of the mean. Lyapunov spectra reveal that f…
We study the nonlinear dynamics of globally coupled nonidentical oscillators in the framework of two order parameter (mean field and amplitude-frequency correlator) reduction. The main result of the paper is the exact solution of the corresponding nonlinear system on an attracting manifold. We present a complete classification of phase portraits and bifurcations, obtain explicit expressions for invariant manifolds (a limit cycle among them) and derive analytical solutions for arbitrary initial data and different regimes.
A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is $\sin θ$, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not $\sin θ$, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.
We study an oscillatory medium with a nonlinear global coupling that gives rise to a harmonic mean-field oscillation with constant amplitude and frequency. Two types of cluster states are found, each undergoing a symmetry-breaking transition towards a related chimera state. We demonstrate that the diffusional coupling is non-essential for these complex dynamics. Furthermore, we investigate localized turbulence and discuss whether it can be categorized as a chimera state.
Universal scaling laws form one of the central issues in physics. A non-standard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we found that a statistical quantity related to fluctuations follows a non-standard scaling law with respect to system size in a synchronized state of globally coupled non-identical phase oscillators [Nishikawa et al., Chaos $\boldsymbol{22}$, 013133 (2012)]. However, it is still unclear how widely this non-standard scaling law is observed. In the present…
The purpose of this paper to analyze in some detail the arguably simplest case of diversity-induced reseonance: that of a system of globally-coupled linear oscillators subjected to a periodic forcing. Diversity appears as the parameters characterizing each oscillator, namely its mass, internal frequency and damping coefficient are drawn from a probability distribution. The main ingredients for the diversity-induced-resonance phenomenon are present in this system as the oscillators display a variability in the individual responses but are induced, by the coupling, to synchronize their responses…
Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviours. Such systems can have chaotic dynamics for $N>4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies showed that some of chaotic attractors in this system are organized by heteroclinic networks. In present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also …
We study a generalization of globally coupled maps, where the elements are updated with probability $p$. When $p$ is below a threshold $p_c$, the collective motion vanishes and the system is the stationary state in the large size limit. We present the linear stability analysis.
We analyze the consequences of symmetry breaking in the coupling in a network of globally coupled identical Stuart-Landau oscillators. We observe that symmetry breaking leads to increased disorderliness in the dynamical behavior of oscillatory states and consequently results in a rich variety of dynamical states. Depending on the strength of the nonisochronicity parameter, we find various dynamical states such as amplitude chimera, amplitude cluster, frequency chimera and frequency cluster states. In addition we also find disparate transition routes to recently observed chimera death state in …
We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models…
We investigate the emergence of chimera and cluster states possessing asymmetric dynamics in globally coupled systems, where the trajectories of oscillators belonging to different subpopulations exhibit different dynamical properties. In an asymmetric chimera state, the trajectory of an element in the synchronized subset is stationary or periodic, while that of an oscillator in the desynchronized subset is chaotic. In an asymmetric cluster state, the periods of the trajectories of elements belonging to different clusters are different. We consider a network of globally coupled chaotic maps as …
The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional description of a system of arbitrary size of identical oscillators coupled all-to-all via their higher-order modes. To demonstrate an application of the formulation, we use a second harmonic globally coupled model, with …
We present a model of identical coupled two-state stochastic units each of which in isolation is governed by a fixed refractory period. The nonlinear coupling between units directly affects the refractory period, which now depends on the global state of the system and can therefore itself become time dependent. At weak coupling the array settles into a quiescent stationary state. Increasing coupling strength leads to a saddle node bifurcation, beyond which the quiescent state coexists with a stable limit cycle of nonlinear coherent oscillations. We explicitly determine the critical coupling co…
In this paper, we propose a dynamic delayed feedback control approach for desynchronization of chaotic-bursting synchronous activities in an ensemble of globally coupled neuronal oscillators. We demonstrate that the difference signal between an ensemble's mean field and its time delayed state, filtered and fed back to the ensemble, can suppress the self-synchronization in the ensemble. These individual units are decoupled and stabilized at the desired desynchronized states while the stimulation signal reduces to the noise level. The effectiveness of the method is illustrated by examples of two…
We investigate a critical exponent related to synchronization transition in globally coupled nonidentical phase oscillators. The critical exponents of susceptibility, correlation time, and correlation size are significant quantities to characterize fluctuations in coupled oscillator systems of large but finite size and understand a universal property of synchronization. These exponents have been identified for the sinusoidal coupling but not fully studied for other coupling schemes. Herein, for a general coupling function including a negative second harmonic term in addition to the sinusoidal …
The prevalence of Milnor attractors has recently been reported in a class of high-dimensional dynamical systems. We study how this prevalence depends on the number of degrees of freedom by using a globally coupled map and show that the basin fraction of Milnor attractors increases drastically around 5-10 degrees of freedom, saturating for higher numbers of degrees of freedom. It is argued that this dominance of Milnor attractors in the basin arises from a combinatorial explosion of the basin boundaries. In addition, the dominance is also found in a system without permutation symmetry, i,e., a …
A Duffing oscillator in a certain parameter range shows period-doubling that shares the same Feigenbaum ratio with the logistic map, which is an important issue in the universality in chaos. In this paper a globally coupled lattice of Duffing flows (GCFL), which is a natural extension of the globally coupled logistic map lattice (GCML), is constructed. It is observed that GCFL inherits various intriguing properties of GCML and that universality at the level of elements is thus lifted to that of systems. Phase diagrams of GCFL are determined, which are essentially the same with those of GCML. S…
We revisit the globally coupled map lattice (GCML). We show that in the so called turbulent regime various periodic cluster attractor states are formed even though the coupling between the maps are very small relative to the non-linearity in the element maps. Most outstanding is a maximally symmetric three cluster attractor in period three motion (MSCA) due to the foliation of the period three window of the element logistic maps. An analytic approach is proposed which explains successfully the systematics of various periodicity manifestations in the turbulent regime. The linear stability of th…
By coupling counter--rotating coupled nonlinear oscillators, we observe a ``mixed'' synchronization between the different dynamical variables of the same system. The phenomenon of amplitude death is also observed. Results for coupled systems with co--rotating coupled oscillators are also presented for a detailed comparison. Results for Landau-Stuart and Rossler oscillators are presented.
The conditions are discussed for which an ensemble of interacting oscillators may demonstrate the Landau-Hopf scenario of successive birth of multi-frequency quasi-periodic motions. A model is proposed that is a network of five globally coupled oscillators characterized by controlled degree of activation of individual oscillators. Illustrations are given for successive birth of tori of increasing dimension via quasi-periodic Hopf bifurcations.
Spatiotemporal chaos in the form of defect-mediated turbulence is known for oscillators coupled by diffusion. Here we explore the same conditions that produce defect turbulence, in an array of oscillators that are coupled through the activator to a regular oscillator. We find that for very small coupling the oscillators behave independent of each other and then there is a transition to complete synchronization. On further increasing the coupling strength, there is period doubling and a transition to chaotic behavior of each driven unit. However the global behavior shows some ordering, and the …
A method to predict the emergence of different kinds of ordered collective behaviors in systems of globally coupled chaotic maps is proposed. The method is based on the analogy between globally coupled maps and a map subjected to an external drive. A vector field which results from this analogy appears to govern the transient evolution of the globally coupled system. General forms of global couplings are considered. Some simple applications are given.
Recently, a novel mixed-synchronization phenomenon is observed in counter-rotating nonlinear coupled oscillators. In mixed-synchronization state: some variables are synchronized in-phase, while others are out-of-phase. We have experimentally verified the occurrence of mixed-synchronization states in coupled counter-rotating chaotic piecewise Rossler oscillator. Analytical discussion on approximate stability analysis and numerical confirmation on the experimentally observed behavior is also given.
We revisit the globally coupled map lattice (GCML). We use a family of universal 'curves of balance' between two conflicting tendencies in the model- the randomness in each map and the coherence due to an averaging interaction - and we locate the posi-nega switch region in the parameter space of the model. We clarify the mechanism of a basic posi-nega switch in the two-cluster regime, which guarantees no mixing of maps across their mean field in the chaotic transient process. We stress that a special attention must be paid for rounding-off errors; otherwise, there fatally occurs an artifitial …
The behaviors of coupled chaotic oscillators before complete synchronization were investigated. We report three phenomena: (1) The emergence of long-time residence of trajectories besides one of the saddle foci; (2) The tendency that orbits of the two oscillators get close becomes faster with increasing the coupling strength; (3) The diffusion of two oscillator's phase difference is first enhanced and then suppressed. There are exact correspondences among these phenomena. The mechanism of these correspondences is explored. These phenomena uncover the route to synchronization of coupled chaotic…
We propose a basic mechanism for isochronal synchrony and communication with mutually delay-coupled chaotic systems. We show that two Ikeda ring oscillators (IROs), mutually coupled with a propagation delay, synchronize isochronally when both are symmetrically driven by a third Ikeda oscillator. This synchronous operation, unstable in the two delay-coupled oscillators alone, facilitates simultaneous, bidirectional communication of messages with chaotic carrier waveforms. This approach to combine both bidirectional and unidirectional coupling represents an application of generalized synchroniza…
The Globally Coupled Map Lattice (GCML) is one of the basic model of the intelligence activity. We report that, in its so-called turbulent regime, periodic windows of the element maps foliate and systematically control the dynamics of the model. We have found various cluster attractors. In one type of them, the maps split into several almost equally populated clusters and the clusters mutually oscillate with a period (p) that is the same with the number of clusters (c). We name them as maximally symmetric cluster attractors (MSCA's). The most outstanding are the p3c3 MSCA and its bifurcate. Th…
We investigate phase synchronization between two identical or detuned response oscillators coupled to a slightly different drive oscillator. Our result is that phase synchronization can occur between response oscillators when they are driven by correlated (but not identical) inputs from the drive oscillator. We call this phenomenon Generalized Phase Synchronization (GPS) and clarify its characteristics using Lyapunov exponents and phase difference plots.
Collective behavior is studied in globally coupled maps with distributed nonlinearity. It is shown that the heterogeneity enhances regularity in the collective dynamics. Low-dimensional quasiperiodic motion is often found for the mean-field, even if each element shows chaotic dynamics. The mechanism of this order is due to the formation of an internal bifurcation structure, and the self-consistent dynamics between the structures and the mean-field. Keywords: Globally Coupled Map with heterogeneity, Collective behavior
We show that universality in chaotic elements can be lifted to that in complex systems. We construct a globally coupled Flow lattice (GCFL), an analog of a globally coupled Map lattice (GCML). We find that Duffing GCFL shows the same behavior with GCML; population ratio between synchronizing clusters acts as a bifurcation parameter. Lorenz GCFL exhibits interesting two quasi-clusters in an opposite phase motion. Each of them looks like Will o' the wisp; they dance around in opposite phase.
The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius--Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The complete bifurcation behaviour of coupled tent maps near the chaotic band merging point is presented. Furthermore the time independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations.
We propose a new method to investigate collective behavior in a network of globally coupled chaotic elements generated by a tent map. In the limit of large system size, the dynamics is described with the nonlinear Frobenius-Perron equation. This equation can be transformed into a simple form by making use of the piecewise linear nature of the individual map. Our method is applied successfully to the analyses of stability of collective stationary states and their bifurcations.
Clustering bifurcations are investigated by considering models of globally coupled map lattices. Typical classes of clustering bifurcations are revealed. The clustering bifurcation thresholds of the coupled system are closely related to the bifurcation structures of single map. In particular, cluster-doubling bifurcation induced period-doubling bifurcations and clustering induced chaos are found. At the onset of multiple-cluster states, equal-site-occupation-partition, and consequently, equal-phase-shift states (the so-called antiphase states reported previously, Phys. Rev. Lett. 65, 1749 (199…
Bifurcations in a system of coupled maps are investigated. Using symbolic dynamics it is proven that for coupled shift maps the well known space--time--mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state. For coupled non--hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour. The anomalous dependence of fluctuations on the system size is attributed to these bifurcations.
For a system of globally coupled chaotic maps with bistable behaviour we relate the rate function for large deviations in the system size at finite time to dynamical properties of the self consistent Perron-Frobenius operator (SCPFO) that describes the system in the infinite size limit.
It is shown how different globally coupled map systems can be analyzed under a common framework by focusing on the dynamics of their respective global coupling functions. We investigate how the functional form of the coupling determines the formation of clusters in a globally coupled map system and the resulting periodicity of the global interaction. The allowed distributions of elements among periodic clusters is also found to depend on the functional form of the coupling. Through the analogy between globally coupled maps and a single driven map, the clustering behavior of the former systems …
The phenomenon of turbulence is investigated in the context of globally coupled maps. The local dynamics is given by the Chaté-Manneville minimal map previously used in studies of spatiotemporal intermittency in locally coupled map lattices. New features arise in the globally coupled system; for instance, the transition to turbulence takes place discontinuously at some critical values of the parameters of the system. The critical boundaries between different regimes (laminar, turbulent and fully turbulent) of the system are calculated on the parameter space. Windows of turbulence are present o…
A mean-field formulation is used to investigate the bifurcation diagram for globally coupled tent maps by means of an analytical approach. It is shown that the period doubling sequence of the single site map induces a continuous family of periodic states in the coupled system. This type of collective motion breaks the ergodicity of the coupled map lattice. The stability analysis suggests that these states are stable for weak coupling strength but opens the possibility for more complicated types of motion in the regime of moderate coupling.
We study a generic model of globally coupled rotors that includes the effects of noise, phase shift in the coupling, and distributions of moments of inertia and natural frequencies of oscillation. As particular cases, the setup includes previously studied Sakaguchi-Kuramoto, Hamiltonian and Brownian mean-field, and Tanaka-Lichtenberg-Oishi and Acebrón-Bonilla-Spigler models. We derive an exact solution of the self-consistent equations for the order parameter in the stationary state, valid for arbitrary parameters in the dynamics, and demonstrate nontrivial phase transitions to synchrony that i…
The relation between relaxation and diffusion is investigated in a Hamiltonian system of globally coupled rotators. Diffusion is anomalous if and only if the system is going towards equilibrium. The anomaly in diffusion is not anomalous diffusion taking a power-type function, but is a transient anomaly due to non-stationarity. Contrary to previous claims, in quasi-stationary states, diffusion can be explained by a stretched exponential correlation function, whose stretching exponent is almost constant and correlation time is linear as functions of degrees of freedom. The full time evolution is…
In assemblies of globally coupled dynamical units, weak noise perturbing independently the individual units can cause anomalous dispersion in the synchronized cloud of the units in the phase space. When the noise-free dynamics of the synchronized assembly is nonperiodic, various moments of the linear dimension of the cloud as a function of the noise strength exhibit multiscaling properties with parameter-dependent scaling exponents. Some numerical evidence of this peculiar behavior as well as its interpretation in terms of a multiplicative stochastic process with small additive noise is provid…
Using a combination of analytical and numerical techniques, we show that chaos in globally-coupled identical dynamical systems, be they dissipative or Hamiltonian, is both extensive and sub-extensive: their spectrum of Lyapunov exponents is asymptotically flat (thus extensive) at the value $λ_0$ given by a single unit forced by the mean-field, but sandwiched between sub-extensive bands containing typically $\mathcal{O}(\log N)$ exponents whose values vary as $λ\simeq λ_\infty + c/\log N$ with $λ_\infty \neq λ_0$.
