Synchronization: Bringing Order to Chaos A. Pikovsky Institut for - - PowerPoint PPT Presentation

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 observed a synchronization of two pendulum clocks

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He described:

“. . . It is quite worths noting that when we suspended two clocks so constructed from two hooks imbedded in the same wooden beam, the motions of each pendulum in opposite swings were so much in agreement that they never receded the least bit from each other and the sound of each was always heard simultaneously. 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 unexpected result, but after a careful examination finally found that the cause of this 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 practice. 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 inevitable small differences.”

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Modern Times

• 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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Biological observations

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

• n some 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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Basic effect

A pendulum clock generates a (nearly) periodic motion characterized by the period T and the frequency ω = 2π

T

α

• α

time period T

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Two such clocks have different periods frequencies

• α

time period T

1

period T2

1 2 1,2

α α

Being coupled, they adjust their rhythms and have the same frequency ω1 < Ω < ω2. There are different possibilities: in-phase and out-of-phase

• ut of phase

in phase

time time

1,2 1,2

α α

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Synchronization occurs within a whole region of parameters

• ∆ ω

synchronization region

∆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 only Self-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 all

variables)

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y x x time

PHASE is the variable proportional to the fraction of the period amplitude measures deviations from the cycle

• amplitude (form) of oscillations is fixed and stable
• PHASE of oscillations is free

A phase amplitude

• 2π

φ

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Examples: amplifier with a feedback loop clocks: pendulum, electronic,...

• utput

Amplifier input Speaker Microphone

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metronom lasers, elsctronic generators, whistle, Josephson junction, spin-torque oscillators

ic c n

M Damping Precession Spin torque ‘Fixed’ layer Spacer ‘Free’ layer x y z H0

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The concept can be extended to non-physical systems!

Ecosystems, predator-pray, rhythmic (e.g. circadian) processes in cells and organizms relaxation integrate-and-fire oscillators

"firing" accumulation threshold level threshold level

water outflow water level

1

t2 t time time T T

e.g., firing neuron

cell potential time

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Autonomous oscillator

• amplitude (form) of oscillations is fixed and stable
• PHASE of oscillations is free

A phase amplitude

• 2π

φ

˙ 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 sin ωt):

• nly 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-periodic If 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 sin ϕ Parameters in the Adler equation (1946): ∆ω = ω0 − ω detuning ε forcing strength

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Solutions of the Adler equation

dϕ dt = ∆ω + ε sin ϕ Fixed point for |∆ω| < ε: Frequency entrainment Ω = ˙ θ = ω Phase locking ϕ = θ − ψ = const Periodic orbit for |∆ω| > ε: an asynchronous quasiperiodic motion ϕ ϕ |∆ω| < ε |∆ω| > ε

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Phase dynamics as a motion of an overdamped particle in an inclined potential dϕ dt = −dU(ϕ) dϕ U(ϕ) = −∆ω · ϕ + ε cos ϕ |∆ω| < ε |∆ω| > ε

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Synchronization region – Arnold tongue

ω ω_0 ε

• ∆ ω

synchronization region

∆Ω

Unusual situation: synchronization occurs for very small force ε → 0, but cannot be obtain with a simple perturbation method: the perturbation theory is singular due to a degeneracy (vanishing Lyapunov exponent)

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More generally: synchronization of higher order is possible, whith a relation Ω

ω = m n

2:1 2 /2 /2 /2 /2 1 1:3 1:2 2:3 1:1 3ω 3ω ω0 ω0 2ω 3ω 3ω ω

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 synchrony quasiperiodicity

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Example: Periodically driven Josephson junction

Synchronization regions – Shapiro steps, frequency ∼ voltage V =

2e ˙

ϕ

[Lab. Nat. de m´ etrologie et d’essais]

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Example: Radio-controlled clocks

Atomic clocks in the PTB, Braunschweig Radio-controlled clocks

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Example: circadian rhythm

asleep Constant conditions 6 12 18 24 6 12 24 18 1 5 10 15 hours Light - dark awake

Jet-lag is the result of the phase difference shift – one needs to resynchronize

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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: dθ1 dt = ω1 dθ2 dt = ω2 Two weakly coupled oscillators: dθ1 dt = ω1 + εG1(θ1, θ2) dθ2 dt = 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 sin ϕ Parameters: ∆ω = ω1 − ω2 detuning ε coupling strength

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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

• x
• scilloscope
• y

Condenser readings 36 40 44 48 52 56 60 Beat frequency

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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 particle in an inclined (∝ ∆ω) potential ρ = mean flow = smooth function of parameters ε > ∆ω ε < ∆ω

with noise without noise

∆ω ∆Ω

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Large noise: no synchronization Small noise: rare irregular phase slips and long phase-locked intervals

200 400 600 800 1000 −30 −20 −10 10 no force small noise large noise

−π π ϕ time Phase difference ϕ

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Phase of a chaotic oscillator

R¨

• ssler attractor:

˙ x = −y − z ˙ y = x + 0.15y ˙ z = 0.4 + z(x − 8.5)

−10 10 −10 10

x y 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)]

5 10 15 20

x

5 10 15

y

5 10 15 20

x

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

10 20 30 20 40

(x2 + y 2)1/2 z

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Phase dynamics in a chaotic oscillator

A model phase equation: dθ

dt = 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

• f

chaos

200 400 600 800

−1 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 = dθ

dt is easy to calculate

(e.g. Ω = 2π limt→∞ Nt/t, N is a number of maxima) Forced R¨

• ssler oscillator:

˙ x = −y − z + E cos(ωt) ˙ y = x + ay ˙ z = 0.4 + z(x − 8.5)

0.9 0.95 1 1.05 1.1 1.15 0.2 0.4 0.6 0.8

• 0.1

0.1 0.2

E ω 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 W (θ, t) has a sharp peak near θ = ωt + const.

−15 −5 5 15

x

−15 −5 5 15

y

−15 −5 5 15

x 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 synchronized cases

−0.01 0.01 −0.01 0.01

I (n) I (n+12) a)

−0.01 0.01 −0.01 0.01

I (n) I (n+12) b)

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Synchronization region:

6900 6950 7000 7050 7100 7150 7200 7250 7300 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Pacer Amplitude (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 different amplitudes of forcing:

• ∆ ω

synchronization region

∆Ω 45 / 68

Synchronization region: ω ω_0 ε

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Unified description of regular, noisy, and chaotic

• scillators
• scillators

autonomous chaotic forced noisy

• scillators

periodic

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Many coupled oscillators

Typical setups: Lattices Networks Global coupling

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Lattice of phase oscillators

dϕn dt = ωn + εq(ϕn+1 − ϕn) + εq(ϕn−1 − ϕn) Small ε: no synchrony Intermediate ε: clusters Large ε: full synchrony (one cluster)

20 40 60 80 100

site k

−2 2

Ωk

−2 2 −2 2

a) b) c)

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Continuous phase profile

In a large system frequencies can be entrained but the phases widely distributed For a continuous phase profile in the case of noisy/chaotic

• scillators 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 dvn dt = 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: dvn dt = 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) dvn dt = ∇d cos v = cos vn+1 − cos vn−1 20 40 60 80 100 120 10 20 30 40 50 60 70 80 time space

π/5

v

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Global coupling of oscillators

Each-to-each coupling ⇐ ⇒ Mean field coupling

1 N 2 3 Σ

X

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Physical systems with global coupling

◮ Josephson junction arrays ◮ Generators (eg spin-torque oscillators, electrochemical

• scillators,...) with a common load

◮ Multimode lasers

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Example: Metronoms on a common plattform

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Example: blinking fireflies

Also electronic fireflies:

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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 0.5 1 100 200 300 400 500 600 700 800 900 1,000

Time (min)

Biopixel GFP Mean 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-all coupling ˙ φi = ωi + ε 1 N

N

• j=1

sin(φj − φi) Can be written as a mean-field coupling ˙ φi = ωi + ε(−X sin φi + Y cos φi) X + iY = M = 1 N

• j

eiφj The natural frequencies are dis- tributed around some mean fre- quency ω0 ω ω0 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 = 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 n(φ)eiφ dφ = Mε π/2

−π/2

g(Mε sin φ) cos φ eiφ dφ

ε εc

|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: R¨

• ssler oscillators with Gaussian distribution of

frequencies ˙ xi = −ωiyi − zi + εX, ˙ yi = ωixi + ayi, ˙ zi = 0.4 + zi(xi − 8.5),

−15 −5 5 15

x

−20 −10 10

y

−20 −10 10

y

(a) (b) (c) (d) −15 −5 5 15

X

−20 −10 10

Y

−20 −10 10

Y 63 / 68

Experiment

Experimental example: synchronization transition in ensemble of 64 chaotic electrochemical oscillators

Kiss, Zhai, and Hudson, Science, 2002

Finite size of the ensemble yields fluctuations of the mean field ∼ 1

N 0.0 0.1 0.2 0.0 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: desynchronization Small each-to-each coupling ⇐ ⇒ coupling via linear mean field

1 N 2 3 Σ X Y

linear unit

Strong each-to-each coupling ⇐ ⇒ coupling via nonlinear mean field

1 N 2 3 Σ X Y

nonlinear unit

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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 the

• scillators are not locked (quasiperiodicity!) establishes

0.2 0.4 0.6 0.8 1 0.5 1 1 1.1

coupling ε

• rder parameter R

ωosc, Ωmf

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Experiment

[Temirbayev et al, PRE, 2013]

Linear coupling Nonlinear coupling

0.2 0.6 1 1 2 0.2 0.4 0.6 0.8 1 1.10 1.12 1.14 1.16 (a) (b) (c) ε R Amin [V] fmf, fi [kHz] 0.2 0.6 1 1 2 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 neural

synchrony at Parkinson [Montaseri et al, Chaos (2013)])

◮ Synchrony in multifrequency populations (eg resonances

ω1 + ω2 = ω3, [Komarov an A.P., PRL (2013)])

◮ Inverse problems: infer coupling function from the signals (eg

ECG + respiration signals yield coupling Heart-Respiration, [Kralemann et al, Nat. Com. (2013)])

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