Classical Kuramoto Model Research Articles

Data clustering based on quantum synchronization

There exists a specific class of methods for data clustering problem inspired by synchronization of coupled oscillators. This approach requires an extension of the classical Kuramoto model to higher dimensions. In this paper, we propose a novel method based on so-called non-Abelian Kuramoto models. These models provide a natural extension of the classical Kuramoto model to the case of abstract particles (called Kuramoto–Lohe oscillators) evolving on matrix Lie groups U(n). We focus on the particular case \(n=2\), yielding the system of matrix ODE’s on SU(2) with the group manifold \(S^3\). This choice implies restriction on the dimension of multivariate data: in our simulations we investigate data sets where data are represented as vectors in \({\mathbb {R}}^k\), with \(k \le 6\). In our approach each object corresponds to one Kuramoto–Lohe oscillator on \(S^3\) and the data are encoded into matrices of their intrinsic frequencies. We assume global (all-to-all) coupling, which allows to greatly reduce computational cost. One important advantage of this approach is that it can be naturally adapted to clustering of multivariate functional data. We present the simulation results for several illustrative data sets.

Solitary states for coupled oscillators with inertia.

Networks of identical oscillators with inertia can display remarkable spatiotemporal patterns in which one or a few oscillators split off from the main synchronized cluster and oscillate with different averaged frequency. Such "solitary states" are impossible for the classical Kuramoto model with sinusoidal coupling. However, if inertia is introduced, these states represent a solid part of the system dynamics, where each solitary state is characterized by the number of isolated oscillators and their disposition in space. We present system parameter regions for the existence of solitary states in the case of local, non-local, and global network couplings and show that they preserve in both thermodynamic and conservative limits. We give evidence that solitary states arise in a homoclinic bifurcation of a saddle-type synchronized state and die eventually in a crisis bifurcation after essential variation of the parameters.

Stochastic Kuramoto oscillators with discrete phase states.

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 model provides a general framework for coupled oscillations in a Markov chain setting.

Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators

We study the emergence of local exponential in an ensemble of generalized heterogeneous Kuramoto oscillators with different intrinsic dynamics. In the classic Kuramoto model, intrinsic dynamics are given by the Kronecker flow with constant natural frequencies. We generalize the constant natural frequencies to smooth functions that depend on the state and time so that it can describe a more realistic situation arising from neuroscience. In this setting, the ensemble of generalized Kuramoto oscillators loses its even when the coupling strength is large. This leads to the study of a concept of relaxed synchronization, which is called synchronization in literature. In this new concept of weak synchronization, the phase diameter of the entire ensemble is uniformly bounded by some constant inversely proportional to the coupling strength. We focus on the complete synchronizability of a subensemble consisting of generalized Kuramoto oscillators with the same intrinsic dynamics; moreover, we provide several sufficient frameworks leading to local exponential of each homogeneous subensemble, although the whole ensemble is not fully synchronized. This is a generalization of an earlier analytical result regarding practical synchronization. We also provide several numerical simulations and compare them with analytical results.

Diversity of coupled oscillators can enhance their synchronization.

The heterogeneity of coupled oscillators is important for the degree of their synchronization. According to the classical Kuramoto model, larger heterogeneity reduces synchronization. Here, we show that in a model for coupled pancreatic β-cells, higher diversity of the cells induces higher synchrony. We find that any system of coupled oscillators that oscillates on two time scales and in which heterogeneity causes a transition from chaotic to damped oscillations on the fast time scale exhibits this property. Thus, synchronization of a subset of oscillating systems can be enhanced by increasing the heterogeneity of the system constituents.

Color Image Segmentation Based on Modified Kuramoto Model

A new approach for color image segmentation is proposed based on Kuramoto model in this paper. Firstly, the classic Kuramoto model which describes a global coupled oscillator network is changed to be one that is locally coupled to simulate the neuron activity in visual cortex and to describe the influence for phase changing by external stimuli. Secondly, a rebuilt method of coupled neuron activities is proposed by introducing and computing instantaneous frequency. Three oscillating curves corresponding to the pixel values of R, G, B for color image are formed by the coupled network and are added up to produce the superposition of oscillation. Finally, color images are segmented according to the synchronization of the oscillating superposition by extracting and checking the frequency of the oscillating curves. The performance is compared with that from other representative segmentation approaches.

Synchronization dynamics in diverse ensemble of noisy phase oscillators with asynchronous phase updates.

Decentralized control of autonomous phase oscillators integrated into networked systems is of great interest for many technological applications, from clock synchronization in sensor nets to coordinated motion in swarm robotics. In the simplest distributed synchronization scheme, each oscillator updates its phase from time to time to a new value equal to the average of its present phase and the phases of its neighbors. Here we describe the resulting synchronization dynamics within a mean-field model where the update actions of different oscillators are completely asynchronous. In particular, it is shown how the steady-state level of synchrony depends on noise intensity and frequency diversity for any given rate of updates. The central part of the analysis is devoted to the case when the correction rate positively correlates with the degree of macroscopic coherence. We demonstrate that depending on relation between correction rate and phase coherence the oscillators may exhibit both continuous and discontinuous transition from incoherence to synchrony upon the change of interaction constant. To illustrate our analytical results, numerical simulations have been performed for a large population of phase oscillators with the proposed type of coupling.

Synchronization transitions in ensembles of noisy oscillators with bi-harmonic coupling

We describe synchronization transitions in an ensemble of globally coupled phase oscillators with a bi-harmonic coupling function, and two sources of disorder—diversity of the intrinsic oscillators’ frequencies, and external independent noise forces. Based on the self-consistent formulation, we derive analytic solutions for different synchronous states. We report on various non-trivial transitions from incoherence to synchrony, with the following possible scenarios: simple supercritical transition (similar to classical Kuramoto model); subcritical transition with large area of bistability of incoherent and synchronous solutions; appearance of a symmetric two-cluster solution which can coexist with the regular synchronous state. We show that the interplay between relatively small white noise and finite-size fluctuations can lead to metastability of the asynchronous solution.

Synchronization of Kuramoto model in a high-dimensional linear space

Abstract In this Letter, Kuramoto model in a high-dimensional linear space is investigated. Some results on the equilibria and synchronization of the classical Kuramoto model are generalized to the high-dimensional Kuramoto model. It is proved that, if the interconnection graph is connected and all the initial states lie in a half part of the state space, the synchronization can be achieved. Finally, numerical simulations are given to validate the obtained theoretical results.

Karhunen–Loève decomposition approach to analyzing complex network synchronization

In this paper, the approach of the Karhunen–Loève decomposition, known also as the proper orthogonal modes (POMs), is taken to analyze phase synchronization of various complex networks with different topologies, namely the classic Kuramoto model, coupled chaotic maps with Gaussian delays and a chain of diffusively coupled bistable oscillators. In the case of the Kuramoto model, the POMs reveal the tendency and the level of synchronization with the increase of the coupling strength for globally coupled networks and scale-free networks, while periodic POMs are found in nearest-neighbor coupled networks. Furthermore, for cluster networks on the Kuramoto model, the first leading POMs based on different time intervals reveal that different sub-groups of nodes synchronize gradually to different levels, eventually leading to the complete phase synchronization. In the case of coupled chaotic maps, some properties of phase synchronization change with the coupling strength value. In the case of the chain of diffusively coupled bistable oscillators, several main POMs not only determine the network phase synchronization but also provide good reconstruction of the network responses.