Low-Dimensional Discrete Kuramoto Model: Hierarchy of Multifrequency Quasiperiodicity Regimes

2014 ◽  
Vol 24 (07) ◽  
pp. 1430022 ◽  
Author(s):  
Alexander P. Kuznetsov ◽  
Yuliya V. Sedova
Keyword(s):  
Coupling Parameter ◽  
Kuramoto Model ◽  
Parameter Plane ◽  
Phase Oscillators ◽  
Repulsive Interactions ◽  
Discrete Phase ◽  
Model Hierarchy ◽  
Frequency Coupling ◽  
Low Dimensional ◽  
Two Parameter

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.

scholarly journals Dynamics in the Sakaguchi-Kuramoto model with bimodal frequency distribution

PLoS ONE ◽  
2020 ◽  
Vol 15 (12) ◽  
pp. e0243196
Author(s):  
Shuangjian Guo ◽  
Yuan Xie ◽  
Qionglin Dai ◽  
Haihong Li ◽  
Junzhong Yang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Natural Frequency ◽  
Frequency Distribution ◽  
Bimodal Distribution ◽  
Kuramoto Model ◽  
Phase Lag ◽  
Phase Oscillators ◽  
Different Types ◽  
Low Dimensional

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.

CHAOTIC ATTRACTOR IN THE KURAMOTO MODEL

2005 ◽  
Vol 15 (11) ◽  
pp. 3457-3466 ◽  
Author(s):  
YURI L. MAISTRENKO ◽  
OLEKSANDR V. POPOVYCH ◽  
PETER A. TASS
Keyword(s):  
Chaotic Attractor ◽  
Nonlinear Dynamical System ◽  
Kuramoto Model ◽  
Phase Oscillators ◽  
Bifurcation And Chaos ◽  
Nonlinear Dynamical ◽  
Transition To Chaos ◽  
Quasiperiodic Dynamics ◽  
The Kuramoto Model ◽  
Low Dimensional

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.

scholarly journals Hysteretic behavior of spatially coupled phase-oscillators

2020 ◽  
Vol 15 ◽  
pp. 18
Author(s):  
Eszter Fehér ◽  
Balázs Havasi-Tóth ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Coupling Parameter ◽  
Kernel Functions ◽  
Kuramoto Model ◽  
Hysteretic Behavior ◽  
Phase Oscillators ◽  
First Order ◽  
Spatial Coupling ◽  
Small Coupling ◽  
Small Kernel ◽  
Order Kernel

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.

scholarly journals A Study on the double crisis vertex in a twoparameter plane*

2002 ◽  
Vol 51 (12) ◽  
pp. 2694
Author(s):  
Hong Ling ◽  
Xu Jian-Xue
Keyword(s):  
Parameter Plane ◽  
Two Parameter

Bifurcation and evolution of a forced and damped Duffing system in two-parameter plane

2018 ◽  
Vol 93 (2) ◽  
pp. 749-766 ◽  
Author(s):  
Jian-Fei Shi ◽  
Yan-Long Zhang ◽  
Xiang-Feng Gou
Keyword(s):  
Parameter Plane ◽  
Duffing System ◽  
Two Parameter

On the Parameter Plane of Nonlinear Coupled Oscillators

1993 ◽  
Vol 48 (5-6) ◽  
pp. 655-662
Author(s):  
Wolfgang Metzler ◽  
Achim Brelle ◽  
Klaus-Dieter Schmidt ◽  
Gerrit Danker ◽  
Matthias Köppe ◽  
...  
Keyword(s):  
Coupled Oscillators ◽  
Nonlinear Oscillator ◽  
Mandelbrot Set ◽  
Parameter Plane ◽  
Two Dimensional ◽  
Routes To Chaos ◽  
Nonlinear Coupled Oscillators ◽  
Two Parameter

Abstract Two well-known bifurcation routes to chaos of two-dimensional coupled logistic maps are embedded in a two-parameter plane of a canonical nonlinear oscillator which contains a non-analytic analogon to the Mandelbrot set.

NONSMOOTH AND SMOOTH BIFURCATIONS IN A DISCONTINUOUS PIECEWISE MAP

2012 ◽  
Vol 22 (05) ◽  
pp. 1250112 ◽  
Author(s):  
ZHIYING QIN ◽  
YUEJING ZHAO ◽  
JICHEN YANG
Keyword(s):  
Fixed Point ◽  
Bifurcation Diagram ◽  
Bifurcation Point ◽  
Peculiar Feature ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Border Collision Bifurcation ◽  
Bifurcation Structures ◽  
Piecewise Map ◽  
Two Parameter

In this paper, a piecewise map with singularity of the power (-1/2) is introduced. For this piecewise map, there is an infinite discontinuous gap on the origin. The conditions of nonsmooth border-collision bifurcation and smooth fold or flip bifurcation are analytically derived. For period-1 fixed point, two-parameter-plane can be divided into seven ranges according to different bifurcation structures. For period-n orbits, codimension-2 bifurcation point may lead to different period-increment sequence, and a peculiar feature is found that there are two different period-increment sequences in the same bifurcation diagram.

scholarly journals Distribution of Order Parameter for Kuramoto Model

2015 ◽  
Vol 3 (9) ◽  
pp. 52-68
Author(s):  
Hui Wu ◽  
Dongwook Kim
Keyword(s):  
Order Parameter ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Kuramoto Model ◽  
Chemical System ◽  
Phase Oscillators ◽  
Numerical Result ◽  
Coupled Nonlinear Oscillators ◽  
Large Populations ◽  
The Kuramoto Model

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.

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