Kuramoto model of
synchronization:equilibrium and
non equilibrium aspects
Stefano Ruffo
Dipartimento di Fisica e Astronomia and CSDC, Università degli Studi di
Firenze and INFN, Italy
Strolling on Chaos, Turbulence and Statistical
Mechanics, Angelo Vulpiani 60th Anniversary,
sept. 22-24, Rome
Center for the Study of Complex Dynamics
University of Florence
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Angelo 1989
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FPU
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Transition to equipartition in FPU
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Computers
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References
S. Gupta, M. Potters and S. Ruffo, One-dimensional lattice of oscillators coupled through
power-law interactions: Continuum limit and dynamics of spatial Fourier modes, Phys. Rev. E, 85,
066201 (2012).
S. Gupta, A. Campa and S. Ruffo, Overdamped dynamics of long-range systems on a
one-dimensional lattice: Dominance of the mean-field mode and phase transition, Phys. Rev. E, 86,
061130 (2012).
S. Gupta, A. Campa and S. Ruffo, Nonequilibrium first-order transition in coupled oscillator
systems with inertia and noise, Phys. Rev. E, 89, 022123 (2014)
S. Gupta, A. Campa and S. Ruffo, Kuramoto model of synchronization: equilibrium and
nonequilibrium aspects, J. Stat. Mech.: Theory and Exp., R08001 (2014)
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Plan
Kuramoto and Sakaguchi models
Equilibrium vs. non equilibrium
First-order phase transition
Complete phase diagram
Linear stability analysis
α-Kuramoto
Zero-mode dominance
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Synchronization
Spontaneous synchronization: Coordination of events to operate a system in unison, in the
absence of any ordering field
Flashing fireflies
Synchronized firing of cardiac pacemaker cells
Phase synchronization in electrical power distribution networks
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The Kuramoto model
A framework to study spontaneous synchronization:
N globally coupled oscillators with distributed natural frequencies
N
dθi
K̃ X
= ωi +
sin(θj − θi )
dt
N j=1
K̃ coupling constant, ωi quenched random variables with distribution g(ω)
P
Order parameter, fraction of phase-locked oscillators: r = |1/N j exp(iθj )|
High K̃: Synchronized phase , r > 0
Low K̃: Incoherent phase, r = 0.
For unimodal g(ω) continuous transition on tuning K̃.
r
i-th
oscillator
1
θi
0
ec
K
e
K
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The Kuramoto-Sakaguchi model
r=0
r 6= 0
0
Ke
Kec
Stochastic fluctuations of the ωi in time
N
dθi
K̃ X
= ωi +
sin(θj − θi ) + ηi (t)
dt
N j=1
< ηi (t) >= 0 , < ηi (t)ηj (t′ ) >= 2Dδij δ(t − t′ )
D
Kec(D)
r=0
0
r 6= 0
Ke
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2nd order dynamics
Two dynamical variables: θi (Phase); vi (Angular velocity)
dθi
= vi
dt
N
dvi
K X
m
= −γvi + ωi +
sin(θj − θi ) + ηi (t)
dt
N j=1
where m is the inertia and γ the friction constant.
Motivation:
An adaptive frequency can explain the slower approach to synchronization observed in
a particular firefly (the Pteropyx mallacae) Ermentrout (1991)
Phase dynamics in electric power distribution networks in the mean-field limit
Filatrella, Nielsen and Pedersen (2008), Rohden, Sorge, Timme
and Witthaut (2012), Olmi and Torcini (2014)
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Previous studies
No noise: Simulations for a Lorentzian g(ω) show a
first-order synchronization transition Tanaka,
Lichtenberg and Oishi (1997)
Analysis in the continuum limit, based on a suitable
Fokker-Planck equation analysis in the limit N → ∞ for
a Lorentzian g(ω): either larger inertia or larger ω
spread makes the system harder to synchronize
Acebron and Spigler (1998); Acebron,
Bonilla and Spigler (2000)
HOWEVER, THE COMPLETE PHASE DIAGRAM HAS
NOT BEEN ADDRESSED
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Rescaling
One can analyze the model in the reduced parameter space (T, σ, m)
dθi
= vi
dt
N
1
1 X
dvi
= − √ vi + σωi +
sin(θj − θi ) + ηi (t)
dt
m
N j=1
where now:
g(ω) has zero average and unit width
< ηi (t)ηj (t′ ) >=
2T
√
δ δ(t
m ij
− t′ )
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Equilibrium and nonequilibrium
The role of σ
σ = 0, ⇒ no external drive ⇒ detailed balance ⇒ equilibrium stationary state
σ > 0 ⇒ non equilibrium stationary state
Continuous transition lines
m = T = 0, σ > 0, Kuramoto
m = 0, T > 0, σ > 0, Sakaguchi, continuous transition, critical line
σ = 0 Hamiltonian system + heat-bath (Brownian Mean Field model), continuous
transition, critical line Chavanis (2013)
σ
Kuramoto model
critical point
r=0
Critical line
r 6= 0
0
r=0
BM
Fc
r 6= 0
riti
cal
line
T
m
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Phase diagram-I
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Phase diagram-II
(b)
coh
0.5 σ inc
σ
σ 0.25
0
T 0.2
0.4
50
25
m
0
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Hysteresis
Order parameter
T=0.25,N=500
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
m=1
m=100
0
σ inc
0.25
Adiabatically tuned σ
0.5
σ coh
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Bistability
For m = 20, T = 0.25, N = 100, and a Gaussian g(ω) with zero mean and unit width, (left)
shows, at σ = 0.195, r vs. time in the stationary state, while (right) shows the distribution
P (r) at several σ’s around 0.195.
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N → ∞ continuum limit
Single-particle distibution f (θ, v, ω, t): Fraction of oscillators at time t and for each ω which
have phase θ and angular velocity v (Periodic in θ and normalized).
Evolution by Kramers equation
”
T ∂2f
∂f
∂f
∂ “ v
= −v
+
,
√ − σω − r sin(ψ − θ) f + √
∂t
∂θ
∂v
m
m ∂v 2
with self-consistent order parameter
r exp(iψ) =
ZZZ
dθdvdωg(ω) exp(iθ)f (θ, v, ω, t)
Homogeneous (r = 0) solution
f inc
√
„
«
(v − σω m)2
1
1
√
exp −
=
2π 2πT
2T
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Linear stability results
Stability analysis gives σ inc : f (θ, v, ω, t) = f inc (θ, v, ω) + eλt δf (θ, v, ω)
∞
X
(−mT )p (1 +
2T
=
emT
p=0
p
mT
)
p!
+∞
Z
−∞
g(ω)dω
1+
p
mT
+ i σω
+
T
λ
√
T m
.
Acebron, Bonilla and Spigler (2000)
The equation for λ has at most one solution with a positive real part and, when the
solution exists, it is necessarily real.
Neutral stability ⇒ λ = 0 gives the stability surface σ inc (m, T ).
Similarly, one can define σ coh (m, T ).
The two surfaces enclose the first-order transition surface σc (m, T ).
Taking proper limits, the surface σ inc (m, T ) meets the critical lines on the (T, σ) and
(m, T ) planes.
The intersection of the surface with the (m, σ) plane gives an implicit formula for
inc
(m, σ).
σnoiseless
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Landau free energy
σ < σ inc
σ = σ inc
(i)
σ inc < σ < σc
(iii)
(ii)
F (r)
σ = σc
(iv)
0
r
1
σc < σ < σ
(v)
11111
00000
00000
11111
coh
σ > σ coh
σ = σ coh
(vi)
(vii)
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Mean-field metastability
∆
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A non equilibrium perspective
Kuramoto model as an overdamped limit of a long-range interacting system.
General dynamics: (1) External drive, (2) Quenched vs. annealed randomness.
No quenched randomness ⇒ Equilibrium
Prob. distr. ∼ exp(−β(K.E. + P.E.)), product measure
Phase transition given by P.E. same for underdamped and overdamped dynamics
Quenched randomness ⇒ Non equilibrium stationary state
Prob. distr. 6= exp(−β(K.E. + P.E.)), not product measure
Dynamics matters: Phase transitions different for underdamped and overdamped
dynamics
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α-Kuramoto
K
dθi
= ωi +
e
dt
N
N
X
j=1,j6=i
sin(θj − θi )
|xj − xi |α
ωi is a quenched random variable with distribution g(ω)
In the continuum limit, the local density ρ(θ; ω, x, t) satisfies the continuity equation
∂ρ
∂ “ ∂θ ”
=−
ρ
∂t
∂θ
∂t
∂θ(ω, x, t)
= ω + κ(α)K
∂t
Z
′
dθ dω
′
′
′ sin(θ − θ)
′
′
′
′
dx
ρ(θ
;
ω
,
x
,
t)g(ω
)
|x′ − x|α
Linear stability analysis of the homogeneous state
ρ(θ; ω, x, t) =
1
+ δρ(θ; ω, x, t)
2π
Dispersion relation
ck (α)K
1−
2
Z
∞
−∞
dω
g(ω)
=0
(λk ± iω)
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Stability of the incoherent state
If g(ω) is symmetric around the mean and non increasing then λk is either real positive or
zero.
The dispersion relation rewrites
1 − ck (α)K
Z
∞
dω
0
λk
g(ω) = 0
2
2
λk + ω
The limit λk → 0 gives the critical couplings
(k)
Kc
(0)
Kc
=
2
ck (α)πg(0)
(1)
< Kc
(2)
< Kc
< ....
and the growth rates λk
2K
(k)
πg(0)Kc
Z
0
∞
hλ i
K λ2 /2
λk
k
k
e
Erfc
g(ω)
=
dω 2
= 1.
√
(k)
λk + ω 2
2
Kc
for Gaussian g(ω).
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Zero-mode dominance
1
r0(t)
r1(t)
r2(t)
r3(t)
0.1
0.01
0.001
0.0001
1e-05
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time t
(0)
α = 0.5 , K = 15 , Kc
(1)
≈ 1.59577 , Kc
(2)
≈ 4.26696 , Kc
(3)
≈ 6.53664 , Kc
≈ 7.71516, . . . ,
so that the Fourier modes 0, 1, 2, 3 are all linearly unstable.
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AU GU RI
AN GELO
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