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!!