Synchronization of phase oscillators in the generalized Sakaguchi-Kuramoto model

2017 ◽  
Vol 118 (6) ◽  
pp. 60005 ◽  
Author(s):  
Yu Xiao ◽  
Wenjing Jia ◽  
Can Xu ◽  
Huaping Lü ◽  
Zhigang Zheng
Author(s):  
Hui Wu ◽  
Dongwook Kim

The synchronization in large populations of interacting oscillators has been observed abundantly in nature, emergining in fields such as physical, biological and chemical system. For this reason, many scientists are seeking to understand the underlying mechansim of the generation of synchronous patterns in oscillatory system. The synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. The Kuramoto model can be used to understand the emergence of synchronization in nextworks of coupled, nonlinear oscillators. In particular, this model presents a phase transition from incoherence to synchronization. In this paper, we investigated the distribution of order parameter γ which describes the strength of synchrony of these oscillators. The larger the order parameter γ is, the more extent the oscillators are synchronized together. This order parameter γ is a critical parameter in the Kuramoto model. Kuramoto gave a initial estimate equation for the value of the order parameter by giving the value of the coupling constant. But our numerical results show that the distribution of the order parameter is slightly different from Kuramoto’s estimation. We gave an estimation for the distribution of order parameter for different values of initial conditions. We discussed how the numerical result will be distributed around Kuramoto’s analytical equation.


2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Huihui Song ◽  
Xuewei Zhang ◽  
Jinjie Wu ◽  
Yanbin Qu

AbstractThis work considers a second-order Kuramoto oscillator network periodically driven at one node to model low-frequency forced oscillations in power grids. The phase fluctuation magnitude at each node and the disturbance propagation in the network are numerically analyzed. The coupling strengths in this work are sufficiently large to ensure the stability of equilibria in the unforced system. It is found that the phase fluctuation is primarily determined by the network structural properties and forcing parameters, not the parameters specific to individual nodes such as power and damping. A new “resonance” phenomenon is observed in which the phase fluctuation magnitudes peak at certain critical coupling strength in the forced system. In the cases of long chain and ring-shaped networks, the Kuramoto model yields an important but somehow counter-intuitive result that the fluctuation magnitude distribution does not necessarily follow a simple attenuating trend along the propagation path and the fluctuation at nodes far from the disturbance source could be stronger than that at the source. These findings are relevant to low-frequency forced oscillations in power grids and will help advance the understanding of their dynamics and mechanisms and improve the detection and mitigation techniques.


2013 ◽  
Vol 25 (11) ◽  
pp. 2934-2975 ◽  
Author(s):  
Renan C. Moioli ◽  
Phil Husbands

The dynamic formation of groups of neurons—neuronal assemblies—is believed to mediate cognitive phenomena at many levels, but their detailed operation and mechanisms of interaction are still to be uncovered. One hypothesis suggests that synchronized oscillations underpin their formation and functioning, with a focus on the temporal structure of neuronal signals. In this context, we investigate neuronal assembly dynamics in two complementary scenarios: the first, a supervised spike pattern classification task, in which noisy variations of a collection of spikes have to be correctly labeled; the second, an unsupervised, minimally cognitive evolutionary robotics tasks, in which an evolved agent has to cope with multiple, possibly conflicting, objectives. In both cases, the more traditional dynamical analysis of the system's variables is paired with information-theoretic techniques in order to get a broader picture of the ongoing interactions with and within the network. The neural network model is inspired by the Kuramoto model of coupled phase oscillators and allows one to fine-tune the network synchronization dynamics and assembly configuration. The experiments explore the computational power, redundancy, and generalization capability of neuronal circuits, demonstrating that performance depends nonlinearly on the number of assemblies and neurons in the network and showing that the framework can be exploited to generate minimally cognitive behaviors, with dynamic assembly formation accounting for varying degrees of stimuli modulation of the sensorimotor interactions.


2010 ◽  
Vol 20 (06) ◽  
pp. 1811-1823 ◽  
Author(s):  
VOLODYMYR MAISTRENKO ◽  
ANNA VASYLENKO ◽  
YURI MAISTRENKO ◽  
ERIK MOSEKILDE

The paper describes the appearance of a novel, high-dimensional chaotic regime, called phase chaos, in a time-discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It arises from the nonlinear interaction among the oscillators, while the individual oscillators behave periodically when left uncoupled. For the four-dimensional time-discrete Kuramoto model, we outline the region of phase chaos in the parameter plane and determine the regions where phase chaos coexists with different periodic attractors. We also study the subcritical frequency-splitting bifurcation at the onset of desynchronization and demonstrate that the transition to phase chaos takes place via a torus destruction process.


2020 ◽  
Vol 30 (11) ◽  
pp. 2175-2227
Author(s):  
Young-Pil Choi ◽  
Jaeseung Lee

We present a new hydrodynamic model for synchronization phenomena which is a type of pressureless Euler system with nonlocal interaction forces. This system can be formally derived from the Kuramoto model with inertia, which is a classical model of interacting phase oscillators widely used to investigate synchronization phenomena, through a kinetic description under the mono-kinetic closure assumption. For the proposed system, we first establish local-in-time existence and uniqueness of classical solutions. For the case of identical natural frequencies, we provide synchronization estimates under suitable assumptions on the initial configurations. We also analyze critical thresholds leading to finite-time blow-up or global-in-time existence of classical solutions. In particular, our proposed model exhibits the finite-time blow-up phenomenon, which is not observed in the classical Kuramoto models, even with a smooth distribution function for natural frequencies. Finally, we numerically investigate synchronization, finite-time blow-up, phase transitions, and hysteresis phenomena.


PLoS ONE ◽  
2020 ◽  
Vol 15 (12) ◽  
pp. e0243196
Author(s):  
Shuangjian Guo ◽  
Yuan Xie ◽  
Qionglin Dai ◽  
Haihong Li ◽  
Junzhong Yang

In this work, we study the Sakaguchi-Kuramoto model with natural frequency following a bimodal distribution. By using Ott-Antonsen ansatz, we reduce the globally coupled phase oscillators to low dimensional coupled ordinary differential equations. For symmetrical bimodal frequency distribution, we analyze the stabilities of the incoherent state and different partial synchronous states. Different types of bifurcations are identified and the effect of the phase lag on the dynamics is investigated. For asymmetrical bimodal frequency distribution, we observe the revival of the incoherent state, and then the conditions for the revival are specified.


2005 ◽  
Vol 15 (11) ◽  
pp. 3457-3466 ◽  
Author(s):  
YURI L. MAISTRENKO ◽  
OLEKSANDR V. POPOVYCH ◽  
PETER A. TASS

The Kuramoto model of globally coupled phase oscillators is an essentially nonlinear dynamical system with a rich dynamics including synchronization and chaos. We study the Kuramoto model from the standpoint of bifurcation and chaos theory of low-dimensional dynamical systems. We find a chaotic attractor in the four-dimensional Kuramoto model and study its origin. The torus destruction scenario is one of the major mechanisms by which chaos arises. L. P. Shilnikov has made decisive contributions to its discovery. We show also that in the Kuramoto model the transition to chaos is in accordance with the torus destruction scenario. We present the general bifurcation diagram containing phase chaos, Cherry flow as well as periodic and quasiperiodic dynamics.


Nonlinearity ◽  
2012 ◽  
Vol 25 (5) ◽  
pp. 1247-1273 ◽  
Author(s):  
Giambattista Giacomin ◽  
Khashayar Pakdaman ◽  
Xavier Pellegrin

2014 ◽  
Vol 24 (07) ◽  
pp. 1430022 ◽  
Author(s):  
Alexander P. Kuznetsov ◽  
Yuliya V. Sedova

The dynamics of a low-dimensional ensemble consisting of a network of five discrete phase oscillators is considered. A two-parameter synchronization picture, which appears instead of the Arnol'd tongues with an increase of the system dimension, is discussed. An appearance of the Arnol'd resonance web is detected on the "frequency–coupling" parameter plane. The cases of attractive and repulsive interactions are discussed.


2020 ◽  
Vol 15 ◽  
pp. 18
Author(s):  
Eszter Fehér ◽  
Balázs Havasi-Tóth ◽  
Tamás Kalmár-Nagy

Motivated by phenomena related to biological systems such as the synchronously flashing swarms of fireflies, we investigate a network of phase oscillators evolving under the generalized Kuramoto model with inertia. A distance-dependent, spatial coupling between the oscillators is considered. Zeroth and first order kernel functions with finite kernel radii were chosen to investigate the effect of local interactions. The hysteretic dynamics of the synchronization depending on the coupling parameter was analyzed for different kernel radii. Numerical investigations demonstrate that (1) locally locked clusters develop for small coupling strength values, (2) the hysteretic behavior vanishes for small kernel radii, (3) the ratio of the kernel radius and the maximal distance between the oscillators characterizes the behavior of the network.


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