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 to the further study on cluster synchronization in various systems.
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 present a linear stability analysis of the incoherent state in a system of globally coupled, identical phase oscillators subject to colored noise. In that we succeed to bridge the extreme time scales between the formerly studied and analytically solvable cases of white noise and quenched random frequencies.
We consider a Kuramoto model of coupled oscillators that includes quenched random interactions of the type used by van Hemmen in his model of spin glasses. The phase diagram is obtained analytically for the case of zero noise and a Lorentzian distribution of the oscillators' natural frequencies. Depending on the size of the attractive and random coupling terms, the system displays four states: complete incoherence, partial synchronization, partial antiphase synchronization, and a mix of antiphase and ordinary synchronization.
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…
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…
In this article we study synchronization of Kuramoto oscillators with heterogeneous frequencies, and where underlying topology is a graph of diameter two. When the coupling strengths between every two connected oscillators are the same, we find an analytic condition that guarantees an existence of a Positively Invariant Set (PIS) and demonstrate that existence of a PIS suffices for frequency synchronization. For graphs of diameter two, this synchronization condition is significantly better than existing general conditions for an arbitrary topology. If the coupling strengths can be different fo…
We study synchronization of coupled Kuramoto oscillators with heterogeneous inherent frequencies and general underlying connectivity. We provide conditions on the coupling strength and the initial phases which guarantee the existence of a Positively Invariant Set (PIS) and lead to synchronization. Unlike previous works that focus only on analytical bounds, here we introduce an optimization approach to provide a computational-analytical bound that can further exploit the particular features of each individual system such as topology and frequency distribution. Examples are provided to illustrat…
Synchronization of coupled oscillators is often described using the Kuramoto model. Here we study a generalization of the Kuramoto model where oscillators communicate with each other through an external medium. This generalized model exhibits interesting new phenomena such as bistability between synchronization and incoherence and a qualitatively new form of synchronization where the external medium exhibits small-amplitude oscillations. We conclude by discussing the relationship of the model to other variations of the Kuramoto model including the Kuramoto model with a bimodal frequency distri…
A knot is a circle embedded in the space. Projecting a knot on a plane, we obtain a diagram which is known as the knot diagram. The vertices of the diagram, where the curved lines are crossed, can be considered as sites occupied by oscillators. The synchronization of these oscillators can be studied by means of a Kuramoto model. Here we propose to define some order parameters, of the complete knot diagram and of its regions, to study the synchronization of the system with regard to the different parts of it.
We study synchronization properties of coupled oscillators on networks that allow description in terms of global mean field coupling. These models generalize the standard Kuramoto-Sakaguchi model, allowing for different contributions of oscillators to the mean field and to different forces from the mean field on oscillators. We present the explicit solutions of self-consistency equations for the amplitude and frequency of the mean field in a parametric form, valid for noise-free and noise-driven oscillators. As an example we consider spatially spreaded oscillators, for which the coupling prope…
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 ($Ω_c$) …
Bursting neurons fire rapid sequences of action potential spikes followed by a quiescent period. The basic dynamical mechanism of bursting is the slow currents that modulate a fast spiking activity caused by rapid ionic currents. Minimal models of bursting neurons must include both effects. We considered one of these models and its relation with a generalized Kuramoto model, thanks to the definition of a geometrical phase for bursting and a corresponding frequency. We considered neuronal networks with different connection topologies and investigated the transition from a non-synchronized to a …
The second-order Kuramoto equation describes synchronization of coupled oscillators with inertia, which occur in power grids for example. Contrary to the first-order Kuramoto equation it's synchronization transition behavior is much less known. In case of Gaussian self-frequencies it is discontinuous, in contrast to the continuous transition for the first-order Kuramoto equation. Here we investigate this transition on large 2d and 3d lattices and provide numerical evidence of hybrid phase transitions, that the oscillator phases $θ_i$, exhibit a crossover, while the frequency spread a real phas…
Synchrony of neuronal ensembles is believed to facilitate information exchange among cortical regions in the human brain. Recently, it has been observed that distant brain areas which are not directly connected by neural links also experience synchronization. Such synchronization between remote regions is sometimes due to the presence of a mediating region connecting them, e.g., \textit{the thalamus}. The underlying network structure of this phenomenon is star-like and motivates us to study the \textit{remote synchronization} of Kuramoto oscillators, {modeling neural dynamics}, coupled by a di…
Synchronizing phase frustrated Kuramoto oscillators, a challenge that has found applications from neuronal networks to the power grid, is an eluding problem, as even small phase-lags cause the oscillators to avoid synchronization. Here we show, constructively, how to strategically select the optimal frequency set, capturing the natural frequencies of all oscillators, for a given network and phase-lags, that will ensure perfect synchronization. We find that high levels of synchronization are sustained in the vicinity of the optimal set, allowing for some level of deviation in the frequencies wi…
We consider the Kuramoto model of globally coupled phase oscillators with time-delayed interactions, that is subject to the Ornstein-Uhlenbeck (Gaussian) colored or the non-Gaussian colored noise. We investigate numerically the interplay between the influences of the finite correlation time of noise $τ$ and the time delay $τ_{d}$ on the onset of the synchronization process. Both cases for identical and nonidentical oscillators had been considered. Among the obtained results for identical oscillators is a large increase of the synchronization threshold as a function of time delay for the colore…
Using a survey of wristwatch synchronization from a randomly selected group of independent volunteers, one can model the system as a Kuramoto-type coupled oscillator network. Based on the phase data, both the order parameter and an estimated value of the coupling is derived and the possibilities for similar research to deduce topology from dynamics are discussed.
Recently, there has been considerable interest in the study of spontaneous synchronization, particularly within the framework of the Kuramoto model. The model comprises oscillators with distributed natural frequencies interacting through a mean-field coupling, and serves as a paradigm to study synchronization. In this review, we put forward a general framework in which we discuss in a unified way known results with more recent developments obtained for a generalized Kuramoto model that includes inertial effects and noise. We describe the model from a different perspective, highlighting the lon…
In the context of the celebrated Kuramoto model of globally-coupled phase oscillators of distributed natural frequencies, which serves as a paradigm to investigate spontaneous collective synchronization in many-body interacting systems, we report on a very rich phase diagram in presence of thermal noise and an additional non-local interaction on a one-dimensional periodic lattice. Remarkably, the phase diagram involves both equilibrium and non-equilibrium phase transitions. In two contrasting limits of the dynamics, we obtain exact analytical results for the phase transitions. These two limits…
The synchronization pattern of a fully connected competing Kuramoto model with a uniform intrinsic frequency distribution $g(ω)$ was recently considered. This competing Kuramoto model assigns two coupling constants with opposite signs, $K_1 < 0$ and $K_2 > 0$, to the $1-p$ and $p$ fractions of nodes, respectively. This model has a rich phase diagram that includes incoherent, $π$, and traveling wave (TW) phases and a hybrid phase transition with abnormal properties that occurs through an intermediate metastable $π$ state. Here, we consider the competing Kuramoto model on Erdős--Rényi (ER) rando…
In a large variety of systems (biological, physical, social etc.), synchronization occurs when different oscillating objects tune their rhythm when they interact with each other. The different underlying network defining the connectivity properties among these objects drives the global dynamics in a complex fashion and affects the global degree of synchrony of the system. Here we study the impact of such types of different network architectures, such as Fully-Connected, Random, Regular ring lattice graph, Small-World and Scale-Free in the global dynamical activity of a system of coupled Kuramo…
Phase turbulence is suppressed by applying common noise additively to the Kuramoto-Sivashinsky type equation, and the noise-induced phase synchronization is realized. The noise strength necessary for the suppression of phase turbulence is evaluated theoretically.
Recently, there has been significant advancement in the machine learning (ML) approach and its application to diverse systems ranging from complex to quantum systems. As one of such systems, a coupled-oscillators system exhibits intriguing collective behaviors, synchronization phase transitions, chaotic behaviors and so on. Even though traditional approaches such as analytical and numerical methods enable to understand diverse properties of such systems, some properties still remain unclear. Here, we applied the ML approach to such systems particularly described by the Kuramoto model, with the…
