synchronisation of coupled oscillators: from huygens
TRANSCRIPT
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Synchronisation of coupledoscillators:
From Huygens clocks to chaoticsystems and large ensembles
A. PikovskyDepartment of Physics, University of Potsdam
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What is synchronization
συν: syn = the same, common χρoνoς: chronos = time
Synchronization: Adjustment of rhythms of oscillating objects due to an
interaction
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Periodic oscillators
Synchronization by external force
Mutual synchronization of two oscillators
Synchronization in oscillatory media
Populations of coupled oscillators
Synchronization by common noise
Chaotic oscillators
Complete/identical synchronization
Phase synchronization
Generalized, master-slave, replica, . . .2
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Historical introduction
Christiaan Huygens (1629-
1695) first observed a syn-
chronization of two pendu-
lum clocks3
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He described:
“. . . It is quite worths noting that when we suspended two clocks so construc-ted from two hooks imbedded in the same wooden beam, the motions of eachpendulum in opposite swings were so much in agreement that they never re-ceded the least bit from each other and the sound of each was always heardsimultaneously. Further, if this agreement was disturbed by some interference,it reestablished itself in a short time. For a long time I was amazed at this un-expected result, but after a careful examination finally found that the cause ofthis is due to the motion of the beam, even though this is hardly perceptible.”
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Lord Rayleigh described synchronization in acoustical systems:
“When two organ-pipes of the same pitch
stand side by side, complications ensue
which not unfrequently give trouble in prac-
tice. In extreme cases the pipes may almost
reduce one another to silence. Even when
the mutual influence is more moderate, it
may still go so far as to cause the pipes to
speak in absolute unison, in spite of inevi-
table small differences.”
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W. H. Eccles and J. H. Vincent applied for a British Patent confirming
their discovery of the synchronization property of a triode generator
Edward Appleton and Balthasar van der Pol extended the experiments
of Eccles and Vincent and made the first step in the theoretical study of
this effect (1922-1927)
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Jean-Jacques Dortous de Mairan reported in 1729 on his experiments
with the haricot bean and found a circadian rhythm (24-hours-rhythm):
motion of leaves continues even without variations of the illuminance
Engelbert Kaempfer wrote after his voyage to Siam in 1680:
“The glowworms . . . represent another shew, which settle on so-
me Trees, like a fiery cloud, with this surprising circumstance, that
a whole swarm of these insects, having taken possession of one
Tree, and spread themselves over its branches, sometimes hide
their Light all at once, and a moment after make it appear again
with the utmost regularity and exactness . . .”.
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Self-sustained oscillators
- generate periodic oscillations without periodic forces
- are dissipative nonlinear systems described by autonomous ODEs
- possess a limit cycle in the phase space
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Laser, periodic chemical reactions, predator-pray system, violine, ...
positive feedback loop
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outputAmplifier
input
Speaker
Microphone
pendulum clock
firing neuron
cell
pot
enti
al
time
metronom
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Autonomous oscillator
- amplitude (form) of oscillations is fixed and stable
- PHASE of oscillations is free due to the time-shift invariance
amplitudephase
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0 2πθ
θ = ω0 (Lyapunov exp. 0)
A = −γ(A−A0) (Lyapunov exp. −γ)
A
θ
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Forced oscillator
With small periodic external force (e.g. ∼ εsinωt):
only the phase θ is affected
dθdt
= ω0 + εG(θ,ψ)dψdt
= ω
ψ is the phase of the external force, G(·, ·) is 2π-periodic
If ω0 ≈ ω then the phase difference ϕ = θ(t)−ψ(t) is slow
⇒ perform averaging by keeping only slow terms (e.g. ∼ sin(θ−ψ))
dϕdt
= Δω+ εsinϕ
Parameters in the Adler equation:Δω = ω0−ω detuning
ε forcing strength11
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Solutions of the Adler equation
dϕdt
= Δω+ εsinϕ
Fixed point for |Δω| < ε:
Frequency entrainment Ω = 〈θ〉 = ωPhase locking ϕ = θ−ψ = const
Periodic orbit for |Δω| > ε: an asynchronous quasiperiodic motion
ϕ ϕ
|Δω| < ε |Δω| > ε
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Phase dynamics as a motion of an overdamped particle in an inclined
potential
dϕdt
= −dU(ϕ)dϕ
U(ϕ) = −Δω ·ϕ+ εcosϕ
|Δω| < ε |Δω| > ε
Allows one to understand what happens in presence of noise:
no perfect locking but phase slips due to excitation over the barrier
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Synchronization region = Arnold tongue
ωω0
ε
Δ ω
synchronizationregion
ΔΩ
Unusual situation: synchronization occurs for very small force ε → 0,
but cannot be obtain with a (linear) perturbation method:
the perturbation theory is singular due to a degeneracy
(vanishing Lyapunov exponent)14
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More generally: synchronization of
higher order is possible, whith a re-
lation Ωω = m
n
2:1
2
/2/2
/2/2
1
1:31:22:31:1
03ω03ωω0 ω0
2ω0 03ω03ωω
0
0
Ω/ω
ω0
2ω
ε
ω
ω
Mathematical description: reduce the system on a torus
dφdt
= ω0 + εG(φ,ψ)dψdt
= ν
to a circle map
φn+1 = φn +ν+ εg(φn)
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φn+1 = φn +ν+ εg(φn)
Rotation number = average number of rotations pro iteration
ρ = limT→∞
12π
φT −φ0T
• ρ = pq rational: stable and unstable periodic orbits
• ρ irrational: quasiperiodic dense filling of the circle
For the continuous-time system: ρ = 〈φ〉〈ψ〉 = ratio of the frequencies
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The simplest ways to observe synchronization:
Lissajous figure
Ω/ω = 1/1 quasiperiodicity Ω/ω = 1/2
forcex
Stroboscopic observation:
Plot phase at each period of forcing
synchronyquasiperiodicity
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Example: circadian rhythm
asleep
Constantconditions
6 12 18 24 6 1224181
5
10
15
hours
Light - dark
awake
The “Jet-Lag” results from the phase shift of the force – a new entrain-
ment takes some time18
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Example: radio-controlled clocks
Atomic clocks at the German institute radio-controlled clock
of standards (PTB) in Braunschweig
⇒19
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Mutual synchronization
Two non-coupled self-sustained oscillators:
dθ1dt
= ω1dθ2dt
= ω2
Two weakly coupled oscillators:
dθ1dt
= ω1 + εG1(θ1,θ2)dθ2dt
= ω2 + εG2(θ1,θ2)
For ω1 ≈ ω2 the phase difference ϕ = θ1−θ2 is slow
⇒ averaging leads to the Adler equation
dϕdt
= Δω+ εsinϕ
Parameters:Δω = ω1−ω2 detuning
ε coupling strength20
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Interaction of two periodic oscillators may be attractive ore repulsive: one
observes in phase or out of phase synchronization, correspondingly
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Example: classical experiments by Appleton
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Condenser readings
36 40 44 48 52 56 60
Bea
t fre
quen
cy
Huygens pendulum clocks: see Bennett, Schatz, Rockwood and Wie-
senfeld, Proc. R. Soc. Lond. A (2002)
Organ pipes: see Abel, Bergweiler and Gerhard-Multhaupt, J. Acoust.
Soc. Am. (2006)
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Synchronization of Josephson junctions
Josephson junction (= pendulum) is a rotator, has a zero Lyapunov ex-
ponent
Voltage = h2e〈θ〉 measures the frequency
3400 3600 3800I (μA)
10
30
50
V1,
V2
(μV
)
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Phase of a chaotic oscillator
Rössler attractor:
x = −y− z
y = x+0.15y
z = 0.4+ z(x−8.5)−10 0 10
−10
0
10
xy
phase should correspond to the zero Lyapunov exponent!
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naive definition of the phase: θ = arctan(y/x)
basing on the Poincaré map:
θ = 2πt − tn
tn+1− tn+2πn tn+1 ≤ t < tn
x
y
θ
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For the topologically simple attractors all definitions are good
Lorenz attractor:
x = 10(y− x)
y = 28x− y− xz
z = −8/3z+ xy0 10 20 30
0
20
40
(x2 + y2)1/2
z
0 2 4 6 8 100
5
10
15
anglePoincare’
time
phas
e/2π
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Phase dynamics in a chaotic oscillator
A model phase equation: dθdt = ω0 +F(A)
(first return time to the surface of section depends on the coordinate on
the surface)
A: chaotic ⇒ phase diffusion ⇒broad spectrum
〈(θ(t)−θ(0)−ω0t)2〉 ∝ Dpt
Dp measures coherence of chaos0 200 400 600 800
−1
0
1
2
time
(θ−
ω0t
)/2π
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dθdt
= ω0 +F(A)
F(A) is like effective noise ⇒
Synchronization of chaotic oscillators ≈≈ synchronization of noisy periodic oscillators ⇒
phase synchronization can be observed while the “amplitudes” remain
chaotic
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Synchronization of a chaotic oscillator by external force
If the phase is well-defined ⇒ Ω = 〈dθdt 〉 is easy to calculate
(e.g. Ω = 2π limt→∞ Nt/t, N is a number of maxima)
Forced
Rössler oscillator:
x = −y− z+E cos(ωt)
y = x+ay
z = 0.4+ z(x−8.5) 0.90.95
11.05
1.11.15 0
0.2
0.4
0.6
0.8
-0.1
0
0.1
0.2
Eω
Ω−ω
phase is locked, amplitude is chaotic
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Stroboscopic observation
Autonomous chaotic oscillator: phases are distributed from 0 to 2π.
Under periodic forcing: if the phase is locked, then the distribution has
a sharp peak near θ = ωt + const.
−15 −5 5 15x
−15
−5
5
15
y
−15 −5 5 15x
synchronized asynchronous
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Phase synchronization of chaotic gas discharge by
periodic pacingTacos et al, Phys. Rev. Lett. 85, 2929 (2000)
Experimental setup:
FIG. 2. Schematic representation of our experimental setup.
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Phase plane projections in non-synchronized and synchronized cases
−0.01 0 0.01
−0.01
0
0.01
I (n)
I (n+
12)
a)
−0.01 0 0.01
−0.01
0
0.01
I (n)I (
n+12
)
b)
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Synchronization region:
6900 6950 7000 7050 7100 7150 7200 7250 73000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Pac
er A
mpl
itude
(V
)
Frequency (Hz) ωω0
ε
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Electrochemical chaotic oscillatorKiss and Hudson, Phys. Rev. E 64, 046215 (2001)
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The synchronized oscillator remains chaotic:
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Frequency difference as a function of driving frequency for different am-
plitudes of forcing:
Δ ω
synchronizationregion
ΔΩ
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Synchronization region:
ωω0
ε
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Unified description of regular, noisy, and chaotic
oscillators
oscillators
autonomous
chaotic
forced
noisy
oscillators
periodic
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Ensembles of globally (all-to-all) couples oscillators
• Physics: arrays of Josephson junctions,
multimode lasers,...
• Biology and neuroscience: cardiac
pacemaker cells, population of fireflies,
neuronal ensembles,...
• Social behavior: applause in a large au-
dience, dance,...
Mutual coupling adjusts phases of indvidual systems, which start to keep
pace with each other
Synchronization appears as
a nonequilibrium order-disorder transition39
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A macroscopic example: Millenium Bridge
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Experiment with Millenium Bridge
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Coupled neurons
Synchronisation in neuronal ensembles is believed to be the reason for
emergence of pathological rhythms in the Parkinson disease and in the
Epilepsy
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Watching synchrony
Watching sinchronous blinking of fireflies has boomed into an industry at
Kuala Selangor firefly park (Malaysia), see www.fireflypark.com
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Kuramoto model: coupled phase oscillators
Phase oscillators with all-to-all coupling (like Adler equation)
φk = ωk + ε1N
N
∑j=1
sin(φ j−φk) = ωk + εK sin(Θ−φk)
System can be written as a mean-field coupling with the mean field
(complex order parameter)
KeiΘ =1N ∑
keiφk
The natural frequencies are distributed
around some mean frequency ω0
≈ finite temperature ωω0
g(ω)
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Synchronisation transition
K
small ε: no synchronization,
phases are distributed uniformly,
mean field = 0
large ε: synchronization, distri-
bution of phases is non-uniform,
mean field = 045
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Theory of transition
Similar to the mean-field theory of ferromagnetic transition:
a self-consistent equation for the mean field
KeiΘ =Z 2π
0n(φ)eiφ dφ = Kε
Z π/2
−π/2g(Kεsinφ)cosφ eiφ dφ
εεc
K
Critical coupling εc ∼ width of distribution g(ω) ∼ “temperature”
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Experiment
Experimental example: synchronization transition
in ensemble of 64 chaotic electrochemical oscillators
Kiss, Zhai, and Hudson, Science, 2002
Finite size of the ensem-
ble yields fluctuations of the
mean field ∼ 1N
0.0 0.1 0.20.0
0.2
0.4
0.6
0.8
1.0
ε
rms(
X)
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Identical oscillators: zero temperature
All frequencies are equal, ε > 0, additional phase shift β in coupling
φk = ω+ ε1N
N
∑j=1
sin(φ j−φk−β) = ω+ εK sin(Θ−φk−β)
Attraction: −π2 < β < π
2 =⇒Synchronization, all phases identical φ1 = . . . = φN = Θ,
maximal order parameter K = 1
Repulsion: −π < β < −π2 and π
2 < β < π =⇒Asynchrony, phases distributed uniformely,
order parameter vanishes K = 048
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Linear vs nonlinear coupling I
• Synchronization of a periodic autonomous oscillator is a nonlinear
phenomenon
• it occurs already for infinitely small forcing
• because the unperturbed system is singular (zero Lyapunov expo-
nent)
In the Kuramoto model “linearity” with respect to forcing is assumed
x = F(x)+ ε1f1(t)+ ε2f2(t)+ · · ·φ = ω+ ε1q1(φ,t)+ ε2q2(φ, t)+ · · ·
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Linear vs nonlinear coupling II
Strong forcing leads to “nonlinear” dependence on the forcing amplitude
x = F(x)+ εf(t)φ = ω+ εq(1)(φ,t)+ ε2q(2)(φ,t)+ · · ·
Nonlineraity of forcing manifests itself in the deformation/skeweness of
the Arnold tongue and in the amplitude depnedence of the phase shift
forcing frequencyforc
ing
ampl
itude
ε
linear nonlinear
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Linear vs nonlinear coupling III
Small each-to-each coupling ⇐⇒ coupling via linear mean field
1 N2 3
ΣX
Y
linearunit
Strong each-to-each coupling ⇐⇒ coupling via nonlinear mean field
1 N2 3
ΣX
Y
nonlinearunit
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Nonlinear coupling: a minimal solvable model
We take the Kuramoto model and assume
nonlinear dependence of coupling strength R and phase shift βon the order parameter K
φk = ω+R(εK)εK sin(Θ−φk +β(εK)) KeiΘ =1N ∑
keiφk
For R = const and β = const the Kuramoto model at zero temperature
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Desynchronization transition: critical couplingφk = ω+R(εK)εK sin(Θ−φk +β(εK))
Synchronous solution φk = Θ and K = 1is stable if −π/2 < β(1,ε) < π/2=⇒ critical coupling is determined by β(1,εq) = ±π/2Transition from attraction to repulsion at a critical coupling strength
Beyond this transition partial synchronization
with 0 < K < 1 is observed
The mean field has frequency Ω = Θ = ωosc = 〈φ〉
This field does not entrain the oscillators
=⇒ quasiperiodic regimes are observed53
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Equations for the phase difference
dφkdt
= ω+R(K,ε)K sin(Θ−φk +β(K,ε))
dΘdt
= Ω
we obtain for ψk = φk−Θ
dψkdt
= ω−Ω+R(K,ε)K sin(β(K,ε)−ψk)
• following Kuramoto: we consider N → ∞ and drop the indices
• from the definition KeiΘ = 〈eiφ〉=⇒ self-consistency condition K = 〈eiψ〉 =
πR−π
eiψρ(ψ)dψ
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Self-consistency condition
• complex equation K =πR
−πeiψρ(ψ)dψ for determination of K and Ω
• probability distribution ρ(ψ) ∼ |ψ|−1 can be explicitely obtained after
normalization
• self-consistent equation yelds
β(K,ε) = ±π/2 Ω = ω±R(K,ε)(1+K2)/2
• with account of this, integration of equation for ψ yelds
ωosc = ω±RK2
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Self-organized quasiperiodicity
• frequencies Ω and ωosc depend on ε in a smooth way
=⇒ generally we observe a quasiperiodicity
• recall the equation for critical coupling β(1,εq) = ±π/2
and compare with just obtained β(K,ε) = ±π/2
• attracting coupling for small mean field
repulsing coupling for large mean field
=⇒ the system sets on exactly on the stabilty border, i.e. in a
self-organized critical state
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Simulation
• non-uniform distribution of oscillator
phases, here for ε− εq = 0.2
• different velocities of oscillators and of
the mean field
Re(Ak)
Im(A
k)
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Example: Josephson junctions
• array of Josephson junctions, shunted by a common RLC-load (capa-
citances are neglected)
h2eR
dφkdt
+ Ic sinφk = I − dQdt
dΦdt
+ rdQdt
+QC
=h2e ∑
k
dφkdt
IJ
R
r C L
• the case of linear RLC-load can be reduced to the Kuramoto model
(Wiesenfeld & Swift, 1995)
• we consider nonlinear load: the magnetic flux Φ = L0Q+L1Q3
• phase equation is of general nonlinear coupling type
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Josephson junctions: numerical results
Critical coupling
εq ≈ 0.130.6
0.8
1
0 0.1 0.2 0.3 0.4 0.54.9
4.92
4.94
4.96K
Ω,〈φ
〉
ε
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General references
• A. Pikovsky, M. Rosenblum, J. Kurths,Synchronization: a universal concept in nonlinear sciencesCambridge University Press (2001)
• S. Strogatz, Sync Hyperion Books (2003)• L. Glass, Synchronization and rhytmic processes in phsyiology, Nature (2001)
Phase synchrony of chaotic oscillators
• A. Pikovsky, Sov. J. Commun. Technol. Electron. (1985)• M. Rosenblum et al, Phys. Rev. Lett. (1996)
Kuramoto model
• Y. Kuramoto, Chemical Oscillations, Waves and TurbulenceSpringer (1984), also Dover edition (2003)
• J. Acebron et al, Rev. Mod. Phys. (2005)• M. Rosenblum and A. Pikovsky, Phys. Rev. Lett. (2007)
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