We examine the onset of synchronization transition in a star network of Kuramoto phase oscillators in the presence of inertia and a time delay in the coupling. A direct correlation between the natural frequencies of the oscillators and their degrees is assumed. The presence of time delay is seen to enhance the onset of first order synchronization. The star network also exhibits different synchronization transitions depending on the value of time delay. An analytical prediction to observe the effect of the time delay is provided and further supported by simulation results. Our findings may help provide valuable insights into the understanding of mechanisms that lead to synchronization on complex networks.
The cluster synchronization is a very important characteristic for the higher harmonic coupling Kuramoto system. A novel transformation is provided, and it gives cluster synchronization by the periodic properties of the density function. The periodic properties of the density function also make the cluster sections' boundaries barrier-like, which helps to explain the sensitiveness of cluster synchronization on the initial conditions of the oscillators. Detailed numerical studies confirm the theoretical predictions from this new view of the symmetry transformation. The work is very beneficial t…
The cluster synchronization (CS) is a very important characteristic for the higher harmonic cou- pling Kuramoto system. A novel method from the symmetry transformation is provided, and it gives CS a profoundly mathematical explanation and clear physical annotation. Detailed numerical studies for the order parameters in various conditions confirm the theoretical predictions from this new view of the symmetry transformation. The work is very beneficial to the further study on CS in various systems.
We study the synchronization of oscillators with inertias and phase shifts, namely the second-order Kuramoto-Sakaguchi model. Using the self-consistent method, we find that the effect of inertia is the introduction of effective phase shifts. The discontinuous synchronization transition of the Kuramoto-Sakaguchi model changes to a continuous one when the value of inertia is small. In addition, we find a new synchronization process, in which with increasing coupling strength the system reaches an oscillating state instead of complete synchronization due to the cross-effect of phase shifts and in…
This paper presents new methods and results on almost global synchronization of coupled Hopf nonlinear oscillators, which are commonly used as the dynamic model of engineered central pattern generators (CPGs). On balanced graphs, any positive coupling gain is proven to induce almost global asymptotic synchronization, and a threshold value for truly global exponential synchronization is also computed. Furthermore, a hierarchical connection between coupled Hopf oscillators and Kuramoto oscillators is identified. Finally, a new result on the synchronization of Kuramoto oscillators with arbitrary …
We propose Moebius maps as a tool to model synchronization phenomena in coupled phase oscillators. Not only does the map provide fast computation of phase synchronization, it also reflects the underlying group structure of the sinusoidally coupled continuous phase dynamics. We study map versions of various known continuous-time collective dynamics, such as the synchronization transition in the Kuramoto-Sakaguchi model of non-identical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and dif…
We study synchronization of Kuramoto oscillators in strongly modular networks in which the structure of the network inside each community is averaged. We find that the dynamics of the interacting communities can be described as an ensemble of coupled planar oscillators. In the limit of a large number of communities, we find a low dimensional description of the level of synchronization between the communities. In this limit, we describe bifurcations between incoherence, local synchrony, and global synchrony. We compare the predictions of this simplified model with simulations of heterogeneous n…
A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function $ω^T L ω$, where $ω$ stands for the vector of the natural frequencies of the oscillators, and $L$ for the network Laplacian matr…
We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles of identical oscillators with the global coupling. In such a setup, with an additional assumption that the initial distribution of oscillators is uniform on the sphere, one can derive equations for order parameters in closed form. The rate of synchronization in a real Kuramoto model depends on the dimension of the sphere. Specifically, synchronization is faster on higher-dimensional spheres. On the other side, real order parameter in complex Kuramoto model…
We study two groups of nonidentical Kuramoto oscillators with differing frequency distributions. Coupling between the groups is repulsive, while coupling between oscillators of the same group is attractive. This asymmetry of interactions leads to an interesting synchronization behavior. For small coupling strength, the mean-fields of both groups resemble a more weakly coupled Kuramoto model. After increasing the coupling strength beyond a threshold, they reach the Bellerophon state of multiple clusters of averagely entrained oscillators. A further increase in the coupling strength then asympto…
Synchronization commonly occurs in many natural and man-made systems, from neurons in the brain to cardiac cells to power grids to Josephson junction arrays. Transitions to or out of synchrony for coupled oscillators depend on several factors, such as individual frequencies, coupling, interaction time delays and network structure-function relation. Here, using a generalized Kuramoto model of time-delay coupled phase oscillators with frequency-weighted coupling, we study the stability of incoherent and coherent states and the transitions to or out of explosive (abrupt, first-order like) phase s…
A system of nearest neighbors Kuramoto-like coupled oscillators placed in a ring is studied above the critical synchronization transition. We find a richness of solutions when the coupling increases, which exists only within a solvability region (SR). We also find that they posses different characteristics, depending on the section of the boundary of the SR where the solutions appear. We study the birth of these solutions and how they evolve when {K} increases, and determine the diagram of solutions in phase space.
We explore both analytically and numerically an ensemble of coupled phase-oscillators governed by a Kuramoto-type system of differential equations. However, we have included the effects of time-delay (due to finite signal-propagation speeds) and network plasticity (via dynamic coupling constants) inspired by the Hebbian learning rule in neuroscience. When time-delay and learning effects combine, novel synchronization phenomena are observed. We investigate the formation of spatio-temporal patterns in both one- and two-dimensional oscillator lattices with periodic boundary conditions and comment…
In this work, we study the synchronization of coupled phase oscillators on the underlying topology of scale-free networks. In particular, we assume that each network's component is an oscillator and that each interacts with the others following the Kuramoto model. We then study the onset of global phase synchronization and fully characterize the system's dynamics. We also found that the resynchronization time of a perturbed node decays as a power law of its connectivity, providing a simple analytical explanation to this interesting behavior.
We investigate the long-time asymptotics of the fluctuation SPDE in the Kuramoto synchronization model. We establish the linear behavior for large time and weak disorder of the quenched limit fluctuations of the empirical measure of the particles around its McKean-Vlasov limit. This is carried out through a spectral analysis of the underlying unbounded evolution operator, using arguments of perturbation of self-adjoint operators and analytic semigroups. We state in particular a Jordan decomposition of the evolution operator which is the key point in order to show that the fluctuations of the d…
A new collective behavior of resonant synchronization is discovered and the ability to retrieve information from brain memory is proposed based on this mechanism. We use modified Kuramoto phase oscillator to simulate the dynamics of a single neuron in self-oscillation state, and investigate the collective responses of a neural network, which is composed of $N$ globally coupled Kuramoto oscillators, to the external stimulus signals in a critical state just below the synchronization threshold of Kuramoto model. The input signals at different driving frequencies, which are used to denote differen…
Using a recently introduced linear reformulation of the Kuramoto model of self-synchronizing oscillator systems (arXiv:0704.1166), we study a new class of analytically solvable oscillator systems defined by a particular coupling scheme. We show that these systems have a logarithimic scaling law in the vicinity of the critical point, which may be seen as anomalous with respect to the usual power-law behavior exhibited by the standard Kuramoto model.
An incorporation of higher-order interactions is known to lead an abrupt first-order transition to synchronization in otherwise smooth second-order one for pair-wise coupled systems. Here, we show that adaptation in higher-order coupling strength may yield completely different phenomena, notably, second-order transition to synchronization and tiered synchronization. Using the Ott-Antonsen approach, we perform rigorous theoretical calculations illustrating the origin of these emerging phenomena along with a complete description of all (un)stable states. Numerical simulations for limited-size ne…
In his classical work, Kuramoto analytically described the onset of synchronization in all-to-all coupled networks of phase oscillators with random intrinsic frequencies. Specifically, he identified a critical value of the coupling strength, at which the incoherent state loses stability and a gradual build-up of coherence begins. Recently, Kuramoto's scenario was shown to hold for a large class of coupled systems on convergent families of deterministic and random graphs. Guided by these results, in the present work, we study several model problems illustrating the link between network topology…
The concept of "remote synchronization" (RS) was introduced in [Phys. Rev. E 85, 026208 (2012)], where synchronization in a star network of Stuart-Landau oscillators was investigated. In the RS regime therein described, the central hub served as a transmitter of information between peripheral nodes, while maintaining independent dynamics that were asynchronous with the rest of the network. One of the key conclusions of that paper was that RS cannot occur in pure phase-oscillator networks. Here, we show that the RS regime can exist in networks of Kuramoto oscillators, and that hub nodes can act…
Synchronization is a ubiquitous phenomenon occurring in social, biological, and technological systems when the internal rhythms of their constituents are adapted to be in unison as a result of their coupling. This natural tendency towards dynamical consensus has spurred a large body of theoretical and experimental research in recent decades. The Kuramoto model constitutes the most studied and paradigmatic framework in which to study synchronization. In particular, it shows how synchronization appears as a phase transition from a dynamically disordered state at some critical value for the coupl…
Cortical regions without direct neuronal connections have been observed to exhibit synchronized dynamics. A recent empirical study has further revealed that such regions that share more common neighbors are more likely to behave coherently. To analytically investigate the underlying mechanisms, we consider that a set of n oscillators, which have no direct connections, are linked through m intermediate oscillators (called mediators), forming a complete bipartite network structure. Modeling the oscillators by the Kuramoto-Sakaguchi model, we rigorously prove that mediated remote synchronization,…
We study the synchronization of Kuramoto oscillators on networks where only a fraction of them is subjected to a periodic external force. When all oscillators receive the external drive the system always synchronize with the periodic force if its intensity is sufficiently large. Our goal is to understand the conditions for global synchronization as a function of the fraction of nodes being forced and how these conditions depend on network topology, strength of internal couplings and intensity of external forcing. Numerical simulations show that the force required to synchronize the network wit…
We investigate transition to synchronization in Sakaguchi-Kuramoto (SK) model on complex networks analytically as well as numerically. Natural frequencies of a percentage ($f$) of higher degree nodes of the network are assumed to be correlated with their degrees and that of the remaining nodes are drawn from some standard distribution namely Lorenz distribution. The effects of variation of $f$ and phase frustration parameter $α$ on transition to synchronization are investigated in detail. Self-consistent equations involving critical coupling strength ($λ_c$) and group angular velocity ($Ω_c$) …
A modified Kuramoto model of synchronization in a finite discrete system of locally coupled oscillators is studied. The model consists of N oscillators with random natural frequencies arranged on a ring. It is shown analytically and numerically that finite-size systems may have many different synchronized stable solutions which are characterised by different values of the winding number. The lower bound for the critical coupling is given, as well as an algorithm for its exact calculation. It is shown that in general phase-locking does not lead to phase coherence in 1D.
We explore the collective phase dynamics of Wien-bridge oscillators coupled resistively. We carefully analyze the behavior of two coupled oscillators, obtaining a transformation from voltage to effective phase. From the phase dynamics we show that the coupling can be quantitatively described by Sakaguchi's modification to the Kuramoto model. We also examine an ensemble of oscillators whose frequencies are taken from a flat distribution within a fixed frequency interval. We characterize in detail the synchronized cluster, its initial formation, as well as its effect on unsynchronized oscillator…
Non-linear oscillator networks have revealed properties as the remote synchronization and the quorum sensing. The remote synchronization, defined as the synchronization of nodes not directly connected by any sequence of synchronized nodes, was found firstly in networks of amplitude oscillators and recently in bipartite delayed networks of phase oscillators. The quorum sensing, a biological information scheme discovered in cell aggregates, has been investigated in amplitude oscillators coupled by a common medium. Implications of such findings are important in technology and biology. We show bot…
Kuramoto model is one of the most prominent models for the synchronization of coupled oscillators. It has long been a research hotspot to understand how natural frequencies, the interaction between oscillators, and network topology determine the onset of synchronization. In this paper, we investigate the critical coupling of Kuramoto oscillators on deterministic dense networks, viewed as a natural generalization from all-to-all networks of identical oscillators. We provide a sufficient condition under which the Kuramoto model with non-identical oscillators has one unique and stable equilibrium…
A fundamental understanding of synchronized behavior in multi-agent systems can be acquired by studying analytically tractable Kuramoto models. However, such models typically diverge from many real systems whose dynamics evolve under non-negligible resource constraints. Here we construct a system of coupled Kuramoto oscillators that consume or produce resources as a function of their oscillation frequency. At high coupling, we observe strongly synchronized dynamics, whereas at low coupling we observe independent oscillator dynamics, as expected from standard Kuramoto models. For intermediate c…
We have extended the study of the Kuramoto model with additive Gaussian noise running on the KKI-18 large human connectome graph. We determined the dynamical behavior of this model by solving it numerically in an assumed homeostatic state, below the synchronization crossover point we determined previously. The de-synchronization duration distributions exhibit power-law tails, characterized by the exponent in the range $1.1 < τ_t < 2$, overlapping the in vivo human brain activity experiments by Palva et al. We show that these scaling results remain valid, by a transformation of the ultra-slow e…
Kuramoto oscillators are widely used to explain collective phenomena in networks of coupled oscillatory units. We show that simple networks of two populations with a generic coupling scheme, where both coupling strengths and phase lags between and within populations are distinct, can exhibit chaotic dynamics as conjectured by Ott and Antonsen [Chaos, 18, 037113 (2008)]. These chaotic mean-field dynamics arise universally across network size, from the continuum limit of infinitely many oscillators down to very small networks with just two oscillators per population. Hence, complicated dynamics …
We study synchronization in a Kuramoto model of globally coupled phase oscillators with a bi-harmonic coupling function, in the thermodynamic limit of large populations. We develop a method for an analytic solution of self-consistent equations describing uniformly rotating complex order parameters, both for single-branch (one possible state of locked oscillators) and multi-branch (two possible values of locked phases) entrainment. We show that synchronous states coexist with the neutrally linearly stable asynchronous regime. The latter has a finite life time for finite ensembles, this time gro…
We show that an introduction of a phase parameter ($α$), with $0 \le α\le π/2$, in the interlayer coupling terms of multiplex networks of Kuramoto oscillators can induce explosive synchronization (ES) in the multiplexed layers. Along with the α values, the hysteresis width is determined by the interlayer coupling strength and the frequency mismatch between the mirror (inter-connected) nodes. A mean-field analysis is performed to support the numerical results. Similar to the earlier works, we find that the suppression of synchronization is accountable for the origin of ES. The robustness of ES …
Explosive synchronization (ES) of coupled oscillators on networks is shown to be originated from existence of correlation between natural frequencies of oscillators and degrees of corresponding nodes. Here, we demonstrate that ES is a generic feature of multiplex network of second-order Kuramoto oscillators and can exist in absence of a frequency-degree correlation. A monoplex network of second-order Kuramoto oscillators bearing homogeneous (heterogeneous) degree-distribution is known to display the first-order (second-order) transition to synchronization. We report that multiplexing of two su…
When higher-order interactions enhance synchronization: the case of the Kuramoto model
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（RIKEN跨学科理论与数学科学中心（iTHEMS））；Department of Systems and Control Engineering, Institute of Science Tokyo（东京科学大学系统与控制工程系）；Research Center for Autonomous Systems Materialogy, Institute of Science Tokyo（东京科学大学自主系统材料学研究中心）；Scuola Superiore Meridionale（南方高等学院）
We consider the problem of global synchronization in a large random network of Kuramoto oscillators where some of them are subject to an external periodically driven force. We explore a recently proposed dimensional reduction approach and introduce an effective two-dimensional description for the problem. From the dimensionally reduced model, we obtain analytical predictions for some critical parameters necessary for the onset of a globally synchronized state in the system. Moreover, the low dimensional model also allows us to introduce an optimization scheme for the problem. Our main conclusi…
Complete Synchronization and its Transition in Higher Harmonic Sakaguchi--Kuramoto Oscillators
CSIR-Central Mechanical Engineering Research Institute（印度中央机械工程技术研究所）；Academy of Scientific and Innovative Research (AcSIR)（科学创新研究学院）；Department of Mathematics, National Institute of Technology（国家技术学院数学系）；Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research（匈牙利科学院能源研究中心技术物理与材料科学研究所）；School of Computer Science, UPES（尤帕迪大学计算机科学学院）；Dhirubhai Ambani University（迪鲁拜·安巴尼大学）
The collective dynamics in populations of magnetic spin torque oscillators (STO) is an intensely studied topic in modern magnetism. Here, we show that arrays of STO coupled via dipolar fields can be modeled using a variant of the Kuramoto model, a well-known mathematical model in non-linear dynamics. By investigating the collective dynamics in arrays of STO we find that the synchronization in such systems is a finite size effect and show that the critical coupling-for a complete synchronized state-scales with the number of oscillators. Using realistic values of the dipolar coupling strength be…
We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let $G=(V,E)$ be a connected graph and $(a_{ij})_{i,j=1}^{n}$ denotes its adjacency matrix. Let the function $f:\mathbb{T}^n \rightarrow \mathbb{R}$ be given by $$ f(θ_1, \dots, θ_n) = \sum_{i,j=1}^{n}{ a_{ij} \cos{(θ_i - θ_j)}}.$$ This function has a global maximum when $θ_i = θ$ for all $1\leq i \leq n$. It is known that if every vertex is connected to at least $μ(n-1)$ other vertices for $μ$ su…
Paths to synchronization in the Kuramoto model with inertia
Korea Institute of Energy Technology（韩国能源技术研究院）；North University of China（中北大学）；Ningbo University of Technology（宁波工程学院）；CNR, Institute of Complex Systems（意大利国家研究委员会复杂系统研究所）
Populations of oscillators are present throughout nature. Very often synchronization is observed in such populations if they are allowed to interact. A paradigmatic model for the study of such phenomena has been the Kuramoto model. However, considering real oscillations are rarely isochronous as a function of energy, it is natural to extend the model by allowing the natural frequencies to vary as a function of some dynamical resource supply. Beyond just accounting for a dynamical supply of resources, however, competition over a \emph{shared} resource supply is important in a variety of biologi…
What happens when the paradigmatic Kuramoto model involving interacting oscillators of distributed natural frequencies and showing spontaneous collective synchronization in the stationary state is subject to random and repeated interruptions of its dynamics with a reset to the initial condition? While resetting to a synchronized state, it may happen between two successive resets that the system desynchronizes, which depends on the duration of the random time interval between the two resets. Here, we unveil how such a protocol of stochastic resetting dramatically modifies the phase diagram of t…
A family of stochastic processes has quasi-cycle oscillations if the oscillations are sustained by noise. For such a family we define a Kuramoto-type coupling of both phase and amplitude processes. We find that synchronization, as measured by the phase-locking index, increases with coupling strength, and appears, for larger network sizes, to have a critical value, at which the network moves relatively abruptly from incoherence to complete synchonization as in Kuramoto couplings of fixed amplitude oscillators. We compare several aspects of the dynamics of unsynchronized and highly synchronized …
We investigate transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto (SK) model on complex networks both analytically and numerically. We analytically derive self-consistent equations for group angular velocity and order parameter for the model in the thermodynamic limit. Using the self-consistent equations we investigate transition to synchronization in SK model on uncorrelated scale-free (SF) and Erdős-Rényi (ER) networks in detail. Depending on the degree distribution exponent ($γ$) of SF networks and phase-frustration parameter, the population undergoes from first order…
We present a generalization of the Kuramoto phase oscillator model in which phases advance in discrete phase increments through Poisson processes, rendering both intrinsic oscillations and coupling inherently stochastic. We study the effects of phase discretization on the synchronization and precision properties of the coupled system both analytically and numerically. Remarkably, many key observables such as the steady-state synchrony and the quality of oscillations show distinct extrema while converging to the classical Kuramoto model in the limit of a continuous phase. The phase-discretized …
The Ott--Antonsen ansatz is a powerful tool to extract the behaviors of coupled phase oscillators, but it imposes a strong restriction on the initial condition. Herein, a systematic extension of the Ott--Antonsen ansatz is proposed to relax the restriction, enabling the systematic approximation of the behavior of a globally coupled phase oscillator system with an arbitrary initial condition. The proposed method is validated on the Kuramoto--Sakaguchi model of identical phase oscillators. The method yields cluster and chimera-like solutions that are not obtained by the conventional ansatz.
