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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
Oscillator synchronization with common noise
Overview
Synchronization
Phase Oscillators in the Kuramoto Model
Chemical oscillators
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Oscillator synchronization with common noise
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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Oscillator synchronization with common noise
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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
-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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
Results
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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
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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Oscillator synchronization with common noise
Phase Oscillators in the Kuramoto Model
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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Oscillator synchronization with common noise
Chemical oscillators
Chemical oscillators
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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Oscillator synchronization with common noise
Chemical oscillators
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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Oscillator synchronization with common noise
Chemical oscillators
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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Oscillator synchronization with common noise
Chemical oscillators
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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Oscillator synchronization with common noise
Chemical oscillators
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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Oscillator synchronization with common noise
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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Oscillator synchronization with common noise
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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