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Synchronization: Bringing Order to Chaos
A. Pikovsky
Institut for Physics and Astronomy, University of Potsdam, Germany
Florence, May 14, 2014
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Historical introduction
Christiaan Huygens(1629-1695) first observeda synchronization of twopendulum clocks
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He described:
“. . . It is quite worths noting that when we suspended twoclocks so constructed from two hooks imbedded in the samewooden beam, the motions of each pendulum in oppositeswings were so much in agreement that they never receded theleast bit from each other and the sound of each was alwaysheard simultaneously. Further, if this agreement was disturbedby some interference, it reestablished itself in a short time. Fora long time I was amazed at this unexpected result, but after acareful examination finally found that the cause of this is dueto the motion of the beam, even though this is hardlyperceptible.”
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Lord Rayleigh described synchronization in acoustical systems:
“When two organ-pipes of the samepitch stand side by side, complicationsensue which not unfrequently givetrouble in practice. In extreme cases thepipes may almost reduce one another tosilence. Even when the mutual influenceis more moderate, it may still go so faras to cause the pipes to speak inabsolute unison, in spite of inevitablesmall differences.”
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Modern Times
W. H. Eccles and J. H. Vincent applied for a British Patentconfirming their discovery of the synchronization property of atriode generatorEdward Appleton and Balthasar van der Pol extended theexperiments of Eccles and Vincent and made the first step in thetheoretical study of this effect (1922-1927)
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Biological observations
Jean-Jacques Dortous de Mairan reported in 1729 on hisexperiments with the haricot bean and found a circadian rhythm(24-hours-rhythm): motion of leaves continues even withoutvariations of the illuminanceEngelbert Kaempfer wrote after his voyage to Siam in 1680:
“The glowworms . . . represent another shew, which settleon some Trees, like a fiery cloud, with this surprisingcircumstance, that a whole swarm of these insects,having taken possession of one Tree, and spreadthemselves over its branches, sometimes hide their Lightall at once, and a moment after make it appear againwith the utmost regularity and exactness . . .”.
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Basic effect
A pendulum clock generates a (nearly) periodic motioncharacterized by the period T and the frequency ω = 2π
T
α
α
time
period T
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Two such clockshave differentperiods frequencies
α
time
period T1
period T21
2
1,2
α
α
Being coupled, they adjust their rhythms and have the samefrequency ω1 < Ω < ω2. There are different possibilities: in-phaseand out-of-phase
out of phasein phase
timetime
1,21,2α α
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Synchronization occurs within a whole region of parameters
∆ ω
synchronizationregion
∆Ω
∆ω – frequency mismatch (difference of natural frequencies)∆Ω – difference of observed frequencies
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Self-sustained oscillations
Synchronization is possible for self-sustained oscillators onlySelf-sustained oscillators- generate periodic oscillations- without periodic forces- are active/dissipative nonlinear systems- are described by autonomous ODEs- are represented by a limit cycle on the phase plane (plane of allvariables)
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y
x
x
timePHASE is the variable proportional to the fraction of the periodamplitude measures deviations from the cycle- amplitude (form) of oscillations is fixed and stable- PHASE of oscillations is free
Aphase
amplitude
0 2π0φ
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Examples:amplifier with a feedback loop clocks: pendulum, electronic,...
outputAmplifier
input
Speaker
Microphone
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metronomlasers, elsctronic generators, whistle,Josephson junction, spin-torque oscillators
electrical current exerts on a magnetic medium is called the spin-transfer
torque. First predicted by theorists in 1996 2), the subsequent demonstration
contact attached to a multilayered magnetic
because they are compatible with existing planar technology, are resistant to radiation
M
Damping
Precession
Spintorque
‘Fixed’ layer Spacer‘Free’ layer
x
y
zH0
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The concept can be extended to non-physical
systems!
Ecosystems, predator-pray, rhythmic (e.g. circadian) processes incells and organizmsrelaxation integrate-and-fire oscillators
"firing"accumulation
threshold levelthreshold level
wat
er o
utfl
oww
ater
leve
l
1
t2t
time
time
T
T
e.g., firing neuron
cell
pot
enti
al
time 14 / 68
Autonomous oscillator
- amplitude (form) of oscillations is fixed and stable- PHASE of oscillations is free
Aphase
amplitude
0 2π0φ
θ = ω0 (Lyapunov Exp. 0)
A = −γ(A− A0) (LE −γ)
A
θ
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Forced oscillator
With small periodic external force (e.g. ∼ ε sinωt):only the phase θ is affected
dθ
dt= ω0 + εG (θ, ψ)
dψ
dt= ω
ψ = ωt is the phase of the external force, G (·, ·) is 2π-periodicIf ω0 ≈ ω then ϕ = θ(t)− ψ(t) is slow⇒ perform averaging by keeping only slow terms (e.g.∼ sin(θ − ψ))
dϕ
dt= ∆ω + ε sinϕ
Parameters in the Adlerequation (1946):
∆ω = ω0 − ω detuningε forcing strength
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Solutions of the Adler equation
dϕ
dt= ∆ω + ε sinϕ
Fixed point for |∆ω| < ε:Frequency entrainment Ω = 〈θ〉 = ωPhase locking ϕ = θ − ψ = constPeriodic orbit for |∆ω| > ε: an asynchronousquasiperiodic motion
ϕ ϕ
|∆ω| < ε |∆ω| > ε
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Phase dynamics as a motion of an overdamped particle in aninclined potential
dϕ
dt= −
dU(ϕ)
dϕU(ϕ) = −∆ω · ϕ+ ε cosϕ
|∆ω| < ε |∆ω| > ε
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Synchronization region – Arnold tongue
ωω_0
ε
∆ ω
synchronizationregion
∆Ω
Unusual situation: synchronization occurs for very small forceε→ 0, but cannot be obtain with a simple perturbation method:the perturbation theory is singular due to a degeneracy (vanishingLyapunov exponent)
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More generally: synchronization of higher order is possible, whith arelation Ω
ω = mn
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
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The simplest ways to observe synchronization:Lissajous figure
Ω/ω = 1/1 quasiperiodicity Ω/ω = 1/2
force
x
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Stroboscopic observation:Plot phase at each period of forcing
synchronyquasiperiodicity
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Example: Periodically driven Josephson junction
Synchronization regions – Shapiro steps, frequency ∼ voltageV = ~
2e ϕ
FIG. 4. Current-voltage characteristic of a detector junction
[Lab. Nat. de metrologie et d’essais]
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Example: Radio-controlled clocks
Atomic clocks in the PTB,Braunschweig
Radio-controlledclocks
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Example: circadian rhythm
asleep
Constantconditions
6 12 18 24 6 1224181
5
10
15
hours
Light - dark
awake
Jet-lag is the result of the phase difference shift – one needs toresynchronize
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One can control re-synchronization (eg for shift-work in space)
[E. Klerman, Brigham and Women’s Hospital, Boston]
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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
dϕ
dt= ∆ω + ε sinϕ
Parameters:∆ω = ω1 − ω2 detuning
ε coupling strength
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Interaction of two periodic oscillators may be attractive orerepulsive: one observes in phase or out of phase synchronization,correspondingly
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Example: classical experiments by Appleton
x
oscilloscope
y
Condenser readings
36 40 44 48 52 56 60
Bea
t fre
quen
cy
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Example: Metronoms
Attracting coupling: synchrony in phase
⇔Repulsive coupling: synchrony out of phase
⇔
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Effect of noise
The Langevin dynamics of the phase =the Langevin dynamics of an overdamped particlein an inclined (∝ ∆ω) potentialρ = mean flow = smooth function of parameters
ε > ∆ω ε < ∆ω
with noise
without noise
∆ω
∆Ω
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Large noise: no synchronizationSmall noise: rare irregular phase slips and long phase-lockedintervals
0 200 400 600 800 1000−30
−20
−10
0
10
no forcesmall noiselarge noise
0−π πϕtime
Phasedifferenceϕ
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Phase of a chaotic oscillator
Rossler 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)More advanced: From the condition of maximally uniform rotation[J. Schwabedal et al, PRE (2012)]
15 10 5 0 5 10 15 20x
20
15
10
5
0
5
10
15
y
15 10 5 0 5 10 15 20x
20
10
0
10
20
30
40
y+z
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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 + xy
0 10 20 300
20
40
(x2 + y2)1/2
z
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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 thecoordinate on the surface)
A: chaotic ⇒ phase diffusion ⇒broad spectrum
〈(θ(t)− θ(0)− ω0t)2〉 ∝ Dpt
Dp measures coherence ofchaos
0 200 400 600 800−1
0
1
2
time(θ
−ω0t)/2π
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dθ
dt= 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)
ForcedRossler 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 to2π.Under periodic forcing: if the phase is locked, then W (θ, t) hasa 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 pacing
[Tacos 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 synchronizedcases
−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
Pa
ce
r A
mp
litu
de
(V
)
Frequency (Hz)
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Electrochemical chaotic oscillator
[Kiss 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 differentamplitudes of forcing:
∆ ω
synchronizationregion
∆Ω
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Synchronization region:
ωω_0
ε
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Unified description of regular, noisy, and chaotic
oscillators
oscillators
autonomous
chaotic
forced
noisy
oscillators
periodic
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Many coupled oscillators
Typical setups:LatticesNetworksGlobal coupling
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Lattice of phase oscillators
dϕn
dt= ωn + εq(ϕn+1 − ϕn) + εq(ϕn−1 − ϕn)
Small ε: no synchronyIntermediate ε: clustersLarge ε: full synchrony (onecluster)
0 20 40 60 80 100site k
−2
0
2Ω
k
−2
0
2
−2
0
2 a)
b)
c)
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Continuous phase profile
In a large system frequencies can be entrained but the phaseswidely distributedFor a continuous phase profile in the case of noisy/chaoticoscillators one obtains a KPZ equation
∂ϕ
∂t= ω(x) + ε∇2ϕ+ β(∇ϕ)2 + ξ(x , t)
Roughening means lack of phase synchrony on large scales
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Dissipative vs conservative coupling
dϕn
dt= ω + q(ϕn+1 − ϕn) + q(ϕn−1 − ϕn)
For the phase difference vn = ϕn+1 − ϕn we get
dvndt
= q(−vn) + q(vn+1)− q(−vn−1)− q(vn)
Odd and even parts of the coupling function:q(v) = ε sin v + α cos v :
dvndt
= α∇d cos v + ε∆d sin v
where ∇d and ∆d are discrete nabla and Laplacian operators
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purely conservative coupling, traveling waves (compactons)
dvndt
= ∇d cos v = cos vn+1 − cos vn−1
20 40 60 80 100 120
0 10
20 30
40 50
60 70
800
time
space
π/5
vv
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Global coupling of oscillators
Each-to-each coupling ⇐⇒ Mean field coupling
1 N2 3
ΣX
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Physical systems with global coupling
Josephson junction arrays
Generators (eg spin-torque oscillators, electrochemicaloscillators,...) with a common load
Multimode lasers
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Example: Metronoms on a common plattform
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Example: blinking fireflies
Also electronicfireflies:
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Genetic “fireflies” [Pridle et al, Nature (2011)]
AHL
LuxR-AHL
luxI aiiA
ndh sfGFP
H2O2
Reporter
Oscillator
Coupling
5,000 cells per biopixel
2.5 million total cells
00
0.5
1
100 200 300 400 500 600 700 800 900 1,000
Time (min)
Biopixel GFP
Mean
00
250
500
100 200 300 400 500 600 700 800 900 1,000
Time (min)
Fluorescence
(arbitrary units)
Biopixel number
a c
b d
Dissolved AHL
H2O2 vapour
Media
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A macroscopic example: Millenium Bridge
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Experiment with Millenium Bridge
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Kuramoto model: coupled phase oscillators
Generalize the Adler equation to an ensemble with all-to-allcoupling
φi = ωi + ε1
N
N∑j=1
sin(φj − φi )
Can be written as a mean-field coupling
φi = ωi + ε(−X sinφi + Y cosφi ) X + iY = M =1
N
∑j
e iφj
The natural frequencies are dis-tributed around some mean fre-quency ω0 ωω0
g(ω)
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Synchronisation transition
M
small ε: no synchronization,phases are distributed uni-formly, mean field = 0
large ε: synchronization, dis-tribution of phases is non-uniform, mean field 6= 0
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Theory of transition
Like the mean-field theory of ferromagnetic transition:a self-consistent equation for the mean field
M =
∫ 2π
0n(φ)e iφ dφ = Mε
∫ π/2
−π/2g(Mε sinφ) cosφ e iφ dφ
εεc
|M|
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Globally coupled chaotic oscillators
Each chaotic oscillator is like a noisy phase oscillator⇒ A regular mean field appears at the critical coupling εcr .Example: Rossler oscillators with Gaussian distribution offrequencies
xi = −ωiyi − zi + εX ,
yi = ωixi + ayi ,
zi = 0.4 + zi(xi − 8.5),
−15 −5 5 15x
−20
−10
0
10
y
−20
−10
0
10
y
(a) (b)
(c) (d)
−15 −5 5 15X
−20
−10
0
10
Y
−20
−10
0
10
Y
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Experiment
Experimental example: synchronization transitionin ensemble of 64 chaotic electrochemical oscillatorsKiss, Zhai, and Hudson, Science, 2002
Finite size of the ensembleyields fluctuations of themean field ∼ 1
N
0.0 0.1 0.20.0
0.2
0.4
0.6
0.8
1.0
ε
rms(X)
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Synchronisation transition at zero temperature
[M. Rosenblum, A. P., PRL (2007) ]Identical oscillators = ”zero temerature”
Attraction: synchronization Repulsion: desynchronizationSmall each-to-each coupling ⇐⇒ coupling via linear mean field
1 N2 3
ΣX
Y
linearunit
Strong each-to-each coupling ⇐⇒ coupling via nonlinear meanfield
1 N2 3
ΣX
Y
nonlinearunit
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Loss of synchrony with increase of coupling
Attraction for small coupling Repulsion at large coupling
A state on the border, where the mean field is finite but theoscillators are not locked (quasiperiodicity!) establishes
0 0.2 0.4 0.6 0.8 10
0.5
1
1
1.1
coupling εorder
param
eter
R ωosc,Ωmf
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Experiment
[Temirbayev et al, PRE, 2013]
Linear coupling Nonlinear coupling
0.2
0.6
1
0
1
2
0 0.2 0.4 0.6 0.8 1
1.10
1.12
1.14
1.16
(a)
(b)
(c)
ε
RA
min
[V]
fm
f,f
i[k
Hz]
0.2
0.6
1
0
1
2
0 0.2 0.4 0.6 0.8 1
1.10
1.12
1.14
1.16
ε
(d)
(e)
(f)
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Other topics
Synchronization by common noise [Braun et al, EPL (2012)]
Control of synchrony (eg for suppression of pathological neuralsynchrony at Parkinson [Montaseri et al, Chaos (2013)])
Synchrony in multifrequency populations (eg resonancesω1 + ω2 = ω3, [Komarov an A.P., PRL (2013)])
Inverse problems: infer coupling function from the signals (egECG + respiration signals yield coupling Heart-Respiration,[Kralemann et al, Nat. Com. (2013)])
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