Exact solution at a transition to frequency synchronization of three coupled phase oscillators

2016 ◽  
Vol 94 (9) ◽  
pp. 808-813 ◽  
Author(s):  
Hassan F. El-Nashar
Keyword(s):  
Numerical Solutions ◽  
Frequency Synchronization ◽  
Critical Coupling ◽  
Phase Oscillators ◽  
Analytic Equation ◽  
Analogous Manner ◽  
Analytic Expressions ◽  
Coupling Strengths ◽  
Complete Frequency ◽  
Coupled Phase Oscillators

A model of three bidirectionally coupled phase oscillators in a ring is studied at the transition to a complete frequency synchronization. Analytic expressions for the critical coupling strengths, at which oscillators synchronize to a common frequency, are obtained. These expressions are determined for cases when the initial oscillators’ frequencies are arranged arbitrarily or they are assigned according to fixed separations. Three unidirectionally coupled phase oscillators are synchronized in an analogous manner to the bidirectional system. This finding allows us to find an analytic equation for the critical coupling strength in the case of the model of the unidirectionally coupled phase oscillators. The bifurcation diagrams show excellent agreement between the analytic formulas and the numerical solutions of the differential equations that describe the models.

Conditions and Linear Stability Analysis at the Transition to Synchronization of Three Coupled Phase Oscillators in a Ring

2017 ◽  
Vol 27 (06) ◽  
pp. 1750095
Author(s):  
Hassan F. El-Nashar
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Periodic Boundary Conditions ◽  
Simple Expression ◽  
Analytic Formula ◽  
Critical Coupling ◽  
Phase Oscillators ◽  
Mathematical Expressions ◽  
Coupled Phase Oscillators

We consider a system of three nonidentical coupled phase oscillators in a ring topology. We explore the conditions that must be satisfied in order to obtain the phases at the transition to a synchrony state. These conditions lead to the correct mathematical expressions of phases that aid to find a simple analytic formula for critical coupling when the oscillators transit to a synchronization state having a common frequency value. The finding of a simple expression for the critical coupling allows us to perform a linear stability analysis at the transition to the synchronization stage. The obtained analytic forms of the eigenvalues show that the three coupled phase oscillators with periodic boundary conditions transit to a synchrony state when a saddle-node bifurcation occurs.

scholarly journals Synchronization of Nonidentical Coupled Phase Oscillators in the Presence of Time Delay and Noise

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Somrita Ray ◽  
Monoj Kumar Sen ◽  
Alendu Baura ◽  
Bidhan Chandra Bag
Keyword(s):  
Time Delay ◽  
Phase Difference ◽  
Coupling Strength ◽  
Deterministic System ◽  
Critical Coupling ◽  
Noise Strength ◽  
Phase Oscillators ◽  
Intrinsic Correlation ◽  
Non Gaussian ◽  
Coupled Phase Oscillators

We have studied in this paper the dynamics of globally coupled phase oscillators having the Lorentzian frequency distribution with zero mean in the presence of both time delay and noise. Noise may be Gaussian or non-Gaussian in characteristics. In the limit of zero noise strength, we find that the critical coupling strength (CCS) increases linearly as a function of time delay. Thus the role of time delay in the dynamics for the deterministic system is qualitatively equivalent to the effect of frequency fluctuations of the phase oscillators by additive white noise in absence of time delay. But for the stochastic model, the critical coupling strength grows nonlinearly with the increase of the time delay. The linear dependence of the critical coupling strength on the noise intensity also changes to become nonlinear due to creation of additional phase difference among the oscillators by the time delay. We find that the creation of phase difference plays an important role in the dynamics of the system when the intrinsic correlation induced by the finite correlation time of the noise is small. We also find that the critical coupling is higher for the non-Gaussian noise compared to the Gaussian one due to higher effective noise strength.

scholarly journals Glassy states and super-relaxation in populations of coupled phase oscillators

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
D. Iatsenko ◽  
P.V.E. McClintock ◽  
A. Stefanovska
Keyword(s):  
Phase Oscillators ◽  
Coupled Phase Oscillators

Nonequilibrium Phenomena in Globally Coupled Phase Oscillators: Noise-Induced Bifurcations, Clustering, and Switching

1997 ◽  
Vol 07 (04) ◽  
pp. 917-922
Author(s):  
Seon Hee Park ◽  
Seunghwan Kim ◽  
Seung Kee Han
Keyword(s):  
Phase Transition ◽  
Noise Intensity ◽  
Multiplicative Noise ◽  
Additive Noise ◽  
Critical Value ◽  
Phase Oscillators ◽  
Deterministic Systems ◽  
Nonequilibrium Phenomena ◽  
Noise Models ◽  
Coupled Phase Oscillators

The Nonequilibrium phenomena in a class of globally coupled phase oscillators systems with multiplicative noise are studied. It is shown that at the critical value of the noise intensity the systems undergo a phase transition and converge to clustered states. We also show that the time delay in the interaction between oscillators gives rise to the switching phenomena of clusters. These phenomena are noise-induced effects which cannot be seen in the deterministic systems or in the simple additive noise models.

scholarly journals Periodic coupling suppresses synchronization in coupled phase oscillators

2018 ◽  
Vol 20 (11) ◽  
pp. 113013 ◽  
Author(s):  
Sansan Li ◽  
Xingang Wang ◽  
Shuguang Guan
Keyword(s):  
Phase Oscillators ◽  
Coupled Phase Oscillators

scholarly journals Origin and scaling of chaos in weakly coupled phase oscillators

2018 ◽  
Vol 97 (1) ◽  
Author(s):  
Mallory Carlu ◽  
Francesco Ginelli ◽  
Antonio Politi
Keyword(s):  
Phase Oscillators ◽  
Weakly Coupled ◽  
Coupled Phase Oscillators
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