Synchronization

The Emerging Science of Spontaneous Order

Flashing of Fireflies

SYNC is Ubiquitous

Synchronization is common in nature:(1) Synchronously flashing fireflies.(2) Crickets that chirp in unison.(3) Electrically synchronous pacemaker cells.(4) Groups of women whose menstrual cycles become mutually synchronized.(5) The spin of the moon synchronizes with its orbit.(6) Coupled laser arrays.(7) Josephson Junction Arrays (Superconducting Qub

it SQ)

Gravity and Tidal Force

In-Phase and Anti-Phase

Pacemaker Cell

Sync Heart Cell of Embryonic Chicks

Pulse Perturbation

Phase resetting

Simulation of Oscillator

0 1dx

S x xdt

When x=1, it fires and resets itself to x=0.

0( ) (1 ).tSx t e

Firing Process

Peskin’s Conjectures and results (1975) For arbitrary initial conditions, the

system approaches a state in which all the oscillators are firing synchronously.

This remains true even when the oscillators are not quite identical.

Only proved for two Identical oscillators with small Coupling and weak dissipation .

.

Synchronization of Pulse-Coupled Biological OscillatorsRenato E. Mirollo and Steven H. Strogatz 1990

The MS model (1) Oscillators are identical with the same period T.

(2) Each oscillator has its phase φ = t – nT, n is an integer.

(3) Each oscillator is characterized by a state variable x.

(4) Each oscillator has the same state function x = f(φ).

Basic Ingredients: without ODE

2

20, 0

df d f

d d

1

' 0, " 0

g f

g g

Dynamics of Two Oscillators

0, 0, (( ) ( 0 ( )) ,) )(h R

Return Map Firing Map

Return MapExistence of Attractive Fixed Point

If

( )R ( ) ( (1 ))h g f

( ) ( ( ))h hR

* *( ) .R

*

*

, ( )

, ( ) .

R

R

How about N>2 ? By requiring all to all coupling and the proc

ess of Absorption (Namely, if two OSCs are synchronous, then they will synchronize forever.). Then N OSCs reduce to N-1.

The system for N OSCs becomes synchronized for arbitrary initial Conditions, except for a set of measure zero.

The Weakness of MS-Model

Identical Oscillators. It is not realistic to have all to all

coupling. Absorption is too strong. (Cheating) Instantaneous response. Absence of Refractory Period. (Time

interval in which a second stimulus cannot lead to a subsequent excitation.)

…

THRESHOLD EFFECTS ( 1993, PRE 49, 2668)

Introducing a threshold The firing Map is

Restriction

is equivalent to a refractory period. Breaking ALL to ALL condition

c

( ) 1 ,

( (1 )), 1 .c

c

h

g f

1

2c

c

Threshold (1)

Threshold (2)

Conclusions The MS- model is a special case,

namely, represents the existence of

refractory period and also breaks the All to All coupling Scheme.

By using Absorption, the system synchronizes for arbitrary initial conditions up to a set of measure zero.

0.c

c

A New Twist on old thinking ( 陳福基 )

Firing map h(φ) = g(ε+f(1-φ))

Return map R(φ) = h(h(φ))

if φ> φ* R(φ) <φ

if φ<φ* R(φ) >φ

Attractive Fixed Point!!

So it is commonly believed that this model

can not be synchronized!

Some Details Considering A and B has different ,A Band

A B A

B

If A fires ,the state of B will be raised .

If B fires ,the state of A will be raised .

h ( g( f(1

h ( g( f(1A A

B

Firing maps

R( ) h (h ( ))A B Return Map

Fixed Point Evasion

There is no fixed point!A&B will fire synchronously for all initial conditions!

Kuramoto Model (Non-identical)

1

sin( )N

ii k i

k

d

dt N

Mean Field solution

1

sin( )

1sin

Ni

i ik

ii

dK

dt

KeN

SYNC of Kuramoto’s model

cK K cK K

cK KcK KcK K

NETWORKS ( Nature 410, 268, Strogatz)

Figure 1 Wiring diagrams for complex networks. a, Food web of Little Rock Lake,Wisconsin, currently the largest food web in the primary literature5. Nodes arefunctionally distinct ‘trophic species’ containing all taxa that share the same set ofpredators and prey. Height indicates trophic level with mostly phytoplankton at thebottom and fishes at the top. Cannibalism is shown with self-loops, and omnivory(feeding on more than one trophic level) is shown by different coloured links toconsumers. (Figure provided by N. D. Martinez). b, New York State electric power grid.Generators and substations are shown as small blue bars. The lines connecting themare transmission lines and transformers. Line thickness and colour indicate thevoltage level: red, 765 kV and 500 kV; brown, 345 kV; green, 230 kV; grey, 138 kVand below. Pink dashed lines are transformers. (Figure provided by J. Thorp andH. Wang). c, A portion of the molecular interaction map for the regulatory networkthat controls the mammalian cell cycle6. Colours indicate different types ofinteractions: black, binding interactions and stoichiometric conversions; red,covalent modifications and gene expression; green, enzyme actions; blue,stimulations and inhibitions. (Reproduced from Fig. 6a in ref. 6, with permission.Figure provided by K. Kohn.)© 2001 Macmillan Magazines Ltd

Localized Synchronization in Two Coupled Nonidentical Semiconductor LasersA. Hohl,1 A. Gavrielides,1 T. Erneux,2 and V. Kovanis1

FIG. 1. Schematic of a system of two nonidentical semiconductorlasers mutually coupled at a distance L used to observelocalized synchronization. We find that the laser whichis pumped at a high level may be forced to entrain to the laserwhich is pumped at a significantly lower level.

VOLUME 78, NUMBER 25 PHY S I CAL REV I EW LETTERS 23 JUNE 1997

Spontaneous  synchronization in a network of limit-cycle oscillators with distributed natural frequencies.

Theory of phase locking of globally coupled laser arrays PRA 52, 4089 (1995) Kourtcha

tov et al

Dynamical Evolution Newton’s equation

Phase Space Trajectory

i ii i

dx dpp F

dt dt

                                                   Phase portrait of the pendulum equation.

General dynamical Equations

( , )ii

dqF q t

dt

More Examples on Phase Space

Phase portrait of a damped pendulum with a torque

.

Periodic solutions correspond to closed curves in the phase plane

Deterministic Chaos Nonlinear Equation Dynamical Instability Sensitive Dependence on Initial

Condition No Long Term Prediction

Chaotic Signal

The Bunimovich stadium is a chaotic dynamical billiard

Lorenz Attractor (1)

dx / dt = a (y - x) a=10,b=28dy / dt = x (b - z) - y c=8dz / dt = xy - c z

Lorenz Attractor (2) These figures — made using ρ=28, σ = 10 and β = 8/3 — show

three time segments of the 3-D evolution of 2 trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10-5 in the x-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.

Time t=1 Time t=2 Time t=3

Chaos Synchronization

Chaotic Laser

SYNC Chaotic Laser Output

Chao and Communication

Thank You!!