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.

TWOFOLD IMPACT OF DELAYED FEEDBACK ON COUPLED OSCILLATORS

2007 ◽  
Vol 17 (07) ◽  
pp. 2517-2530 ◽  
Author(s):  
OLEKSANDR V. POPOVYCH ◽  
VALERII KRACHKOVSKYI ◽  
PETER A. TASS
Keyword(s):  
Time Delay ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Phase Synchronization ◽  
Threshold Value ◽  
Phase Oscillators ◽  
Intermediate Coupling ◽  
Rich Variety ◽  
Dynamical Regimes ◽  
The Impact

We present a detailed bifurcation analysis of desynchronization transitions in a system of two coupled phase oscillators with delay. The coupling between the oscillators combines a delayed self-feedback of each oscillator with an instantaneous mutual interaction. The delayed self-feedback leads to a rich variety of dynamical regimes, ranging from phase-locked and periodically modulated synchronized states to chaotic phase synchronization and desynchronization. We show that an increase of the coupling strength between oscillators may lead to a loss of synchronization. Intriguingly, the delay has a twofold influence on the oscillations: synchronizing for small and intermediate coupling strength and desynchronizing if the coupling strength exceeds a certain threshold value. We show that the desynchronization transition has the form of a crisis bifurcation of a chaotic attractor of chaotic phase synchronization. This study contributes to a better understanding of the impact of time delay on interacting oscillators.

Collective dynamics of time-delay-coupled phase oscillators in a frustrated geometry

2017 ◽  
Vol 95 (1) ◽  
Author(s):  
Bhumika Thakur ◽  
Devendra Sharma ◽  
Abhijit Sen ◽  
George L. Johnston
Keyword(s):  
Time Delay ◽  
Phase Oscillators ◽  
Collective Dynamics ◽  
Coupled Phase Oscillators

scholarly journals Multistability in a star network of Kuramoto-type oscillators with synaptic plasticity

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Irmantas Ratas ◽  
Kestutis Pyragas ◽  
Peter A. Tass
Keyword(s):  
Synaptic Plasticity ◽  
Time Scales ◽  
Phase Difference ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Analytical Approach ◽  
Boundary Function ◽  
Phase Oscillators ◽  
Star Network ◽  
Small Parameters

AbstractWe analyze multistability in a star-type network of phase oscillators with coupling weights governed by phase-difference-dependent plasticity. It is shown that a network with N leaves can evolve into $$2^N$$ 2 N various asymptotic states, characterized by different values of the coupling strength between the hub and the leaves. Starting from the simple case of two coupled oscillators, we develop an analytical approach based on two small parameters $$\varepsilon$$ ε and $$\mu$$ μ , where $$\varepsilon$$ ε is the ratio of the time scales of the phase variables and synaptic weights, and $$\mu$$ μ defines the sharpness of the plasticity boundary function. The limit $$\mu \rightarrow 0$$ μ → 0 corresponds to a hard boundary. The analytical results obtained on the model of two oscillators are generalized for multi-leaf star networks. Multistability with $$2^N$$ 2 N various asymptotic states is numerically demonstrated for one-, two-, three- and nine-leaf star-type networks.

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 Estimation of the transmission time of stimulus-locked responses: modelling and stochastic phase resetting analysis

2005 ◽  
Vol 360 (1457) ◽  
pp. 995-999 ◽  
Author(s):  
Peter A Tass
Keyword(s):  
Phase Difference ◽  
Time Difference ◽  
The Other ◽  
Transmission Time ◽  
Phase Resetting ◽  
Phase Oscillators ◽  
Random Forces ◽  
The Cross ◽  
The Difference ◽  
Coupled Phase Oscillators

A model of two coupled phase oscillators is studied, where both oscillators are subject to random forces but only one oscillator is repetitively stimulated with a pulsatile stimulus. A pulse causes a reset, which is transmitted to the other oscillator via the coupling. The transmission time of the cross-trial (CT) averaged responses, i.e. the difference in time between the maxima of the CT averaged responses of both oscillators differs from the time difference between the maxima of the oscillators' resets. In fact, the transmission time of the CT averaged responses directly corresponds to the phase difference in the stable synchronized state with integer multiples of the oscillators' mean period added to it. With CT averaged responses it is impossible to reliably estimate the time elapsing, owing to the stimulus' action being transmitted between the two oscillators.

scholarly journals Dynamical analysis of a stochastic rumor-spreading model with Holling II functional response function and time delay

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Liang’an Huo ◽  
Xiaomin Chen
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Functional Response ◽  
Response Function ◽  
Rapid Development ◽  
Dynamical Behavior ◽  
Deterministic System ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Holling Ii Functional Response

AbstractWith the rapid development of information society, rumor plays an increasingly crucial part in social communication, and its spreading has a significant impact on human life. In this paper, a stochastic rumor-spreading model with Holling II functional response function considering the existence of time delay and the disturbance of white noise is proposed. Firstly, the existence of a unique global positive solution of the model is studied. Then the asymptotic behavior of the global solution around the rumor-free and rumor-local equilibrium nodes of the deterministic system is discussed. Finally, through some numerical results, the validity and availability of theoretical analysis is verified powerfully, and it shows that some factors such as the transmission rate, the intensity of white noise, and the time delay have significant relationship with the dynamical behavior of rumor spreading.

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