Oscillator synchronization with common noise€¦ · Theoretische Physik VI, Statistische Physik d. Nichtgleichgewichts ... Synchronization Phase Oscillators in the Kuramoto Model

Oscillator synchronization with common noise

Kisa Barkemeyer, Zoi Kourkouraidou, Melissa Skarmeta

18 July 2018

Theoretische Physik VI, Statistische Physik d. NichtgleichgewichtsSS 2018

Prof. Dr. Engel, Dr. Totz


Overview

Synchronization

Phase Oscillators in the Kuramoto Model

Chemical oscillators

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Synchronization

Synchronization in natureSynchronization of coupled oscillators plays an important role inmany different areas in nature, e.g. in

I the circadian clock (Hall, Rosbash, Young: Nobel prize 2017)

I neuronal networks

I swarms of birds, fish, and fireflies

Robin Meier & Andre Gwerder: Synchronicity (Thailand), 2015

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Synchronization

Reasons for synchronization

Synchronization can be caused by different mechanisms:

I couplingI attractive → synchronizationI repulsive → desynchronization

I (weak) common noise → synchronization

Interplay of repulsive coupling and common noise→ non-trivial effects

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The Kuramoto Model

Infinitely many phase oscillators + Kuramoto-Sakaguchi coupling+ common noisediscrete representation of phase-dynamics:

ϕi = Ωi + µ

N∑j=1j 6=i

sin(ϕj − ϕi − β) + σξ sin(ϕi )

Ωi natural frequency with distribution g(Ω) = γ

π(γ2+(Ω−Ω0)2)µ couplingβ phase frustration parameterξ(t) gaussian white noise with strength σ

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Model Reduction

I N →∞: introduce probability density function ω(ϕ, t,Ω)

following ∂ω∂t + ∂(ω·v)

∂ϕ = 0

reduce the model according to the Ott-Antonsen ansatz:

I introduction of complex order parameter z(t), |z(t)| ≤ 1:

z(t) = Re iΦ =

∫ ∞−∞

dΩg(Ω)

∫ 2π

0dϕe iϕ ω(ϕ, t,Ω),

I describe ω by its Fourier series, using ωn(Ω, t) = a(Ω, t)n:

ω(ϕ, t,Ω) =g(Ω)

2π

(1 +

∞∑n=1

(a(Ω, t)n e inϕ + a∗(Ω, t)n e−inϕ

))

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-Π - Π

2Π

Π

2

4

8

12ΩHΦ,W,tL

0.96

0.88

0.8

0.72

0.64

0.56

0.48

0.4

0.32

0.24

0.16

0.08

0

Figure: probability density function ω(ϕ, t,Ω) as a function of ϕ (x-axis) andof a (→colored lines), which ranges from 0 (→horizontal line) to 1 (peak).

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Reduced Equations

I deduce a special condition for a(Ω, t):

a (Ω, t) = −iΩ a +σξ

2

(a2 − 1

)+µ

2

(z∗ e iβ − a2 z e−iβ

),

that can be inserted into z(t).

I finally one obtains following equation of motion:

ϕ = Ω + σξ(t) sin(ϕ) + µR sin(Φ− ϕ− β)

= Ω + Im(H(t)e−iϕ

),

with H(t) = µRe−iβe iΦ − σξ(t)

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Dynamics of the order parameter

Transformation of the order parameter R → J = R2/(1− R2

)R ∈ [0, 1] , J ∈ [ 0,∞ [

where R = 0/J = 0⇔ asynchrony, R = 1/J →∞⇔ synchrony

J = µβJ − 2γJ(J + 1)− σξ(t)√

J(J + 1) cos(Φ)

Φ = Ω0 − µ sin(β)J + 1/2

J + 1+ σξ(t)

J + 1/2√J(J + 1)

sin(Φ)

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Stability of the synchronous state for J 1

J = µβJ − 2γJ2 − σξ(t)J cos(Φ)

Φ = Ωµ,0 + σξ(t) sin(Φ)

Average growth rate of J for γ = 0 (identical oscillators):

λ0 = µβ︸︷︷︸≶0

+σ2⟨sin2(Φ)

⟩︸︷︷︸>0

⇒ perfect synchrony possible

Average growth rate of J for γ 6= 0 (non-identical oscillators):

λγ = µβ − 2γ 〈J〉+ σ2⟨sin2(Φ)

⟩⇒ perfect synchrony impossible

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Results

Figure: Dynamics of the frequency differenceϕi−ϕj

2π of two oscillators i , jin an ensemble for attractive (µ = 0.25), repulsive (µ = −0.25) or nocoupling (µ = 0).

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Results

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Influence of coupling and noise on the synchronization ofnon-identical oscillators

I coupling without noise:

I attraction of frequencies for positive coupling (µ > 0)I no effect for repulsive coupling (µ < 0)

I noise without coupling: no influence on the frequencies

I coupling together with noise:I µ > 0: same as without noiseI µ < 0: dispersion of frequencies

→ counterintuitive !

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direct simulation of the population (N=21)

simulation of the Ott-Antonsen-equations

Figure: adapted from Figure 3. in : A. V. Pimenova, D.S. Goldobin, M.Rosenblum and A. Pikovsky, Interplay of coupling and common noise atthe transition to synchrony in oscillator populations, Scientific Reports38518 (2016)

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Mixture of reacting chemical compounds, in which theconcentration of one or more components exhibits periodicchanges

I non-equillibrium thermodynamics

I state of chemical oscillators described by intensity I

I intensity depends on concentration→influences grey value g of each oscillator

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Coupled chemical oscillatorsI Setup:

Figure: adapted: Fig. 1. in : J.F. Totz, J. Rode, M.R. Tinsley, K. Showalter, H. Engel, Spiral wavechimera states in large populations of coupled chemical oscillators, Nature Physics Vol.14, 282-5, 2018

I population of asynchronous chemical oscillatorsI placed in a catalyst-free Belousov-Zhabotinsky (BZ) solutionI coupling: through light, non-local

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Coupled chemical oscillators

Figure: adapted from: J.F. Totz, J. Rode, M.R. Tinsley, K. Showalterand H. Engel, Spiral wave chimera states in large populations of coupledchemical oscillators, Nature Physics Vol.14, 282-285 (2018)

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Model

I oscillators (at position (j , k)) arranged on a 2D-grid:

Ij,k = I0 + K

j+l∑m=j−l

k+l∑n=k−l

e−κr (gm,n(t − τ) − gj,k(t))

K coupling strength, κ coupling range

e−κr non-local coupling kernel, r =√

(m − j)2 + (n − k)2

I0 intensity of the background illuminationτ time-delay ⇔ phase-frustration parameter

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Implementation of noise

similarly to presented model in Ref.1:

I global coupling

I multiplicative noise

Ii = I0 + µ

N∑j=1

(gj(t)− gi (t)) + σ ξ(t) f (gi )

µ coupling strength σ noise intensityξ(t) gaussian noise f (gi ) function of grey value gi

1S. Goldobin, A. V. Pimenova, M. Rosenblum and A. Pikovsky, Competing influence of common noise and

desynchronizing coupling on synchronization in the Kuramoto-Sakaguchi ensemble, Eur. Phys. J. Special Topics226, 1921-1937 (2017)

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Literature

E. Ott and T.M. Antonsen, Low dimensional behavior of largesystems of globally coupled oscillators, Chaos 18, 3, 037113 (2008)

E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott, P. So and T.M.Antonsen, Exact results for the Kuramoto model with a bimodalfrequency distribution, Phys. Rev., E 79, 026204 (2009)

D. M. Abrams, R.E. Mirollo, S.H. Strogatz and D.A. Wiley, SolvableModel for Chimera States of Coupled Oscillators, Phys. Rev. Lett.101, 8, 084103 (2008)

D.S. Goldobin, A. V. Pimenova, M. Rosenblum and A. Pikovsky,Competing influence of common noise and desynchronizing couplingon synchronization in the Kuramoto-Sakaguchi ensemble, Eur. Phys.J. Special Topics 226, 1921-1937 (2017)

D.S. Goldobin, A. V. Pimenova, M. Rosenblum and A. Pikovsky,Interplay of coupling and common noise at the transition tosynchrony in oscillator populations, Nature Scientific Reports 38518(2016)

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Literature

J.F. Totz, J. Rode, M.R. Tinsley, K. Showalter and H. Engel, Spiralwave chimera states in large populations of coupled chemicaloscillators, Nature Physics Vol.14, 282-285 (2018)

J. F. Totz., Synchronization and Waves in Confined Complex ActiveMedia, Ph.D. thesis, TU Berlin, Berlin (2017)

Thank you for your attention!

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Documents

Oscillator synchronization with common noise€¦ · Theoretische Physik VI, Statistische Physik d. Nichtgleichgewichts ... Synchronization Phase Oscillators in the Kuramoto Model