Synchronization in Real Optical Networks
Chaos, Communication and Chimeras in the Laboratory
Rajarshi Roy
Institute for Physical Science and Technology
Department of Physics
Institute for Research in Electronics and Applied
Physics
University of Maryland
College Park, MD
USA
June 20, 2012
Outline
dynamics on an interacting complex network of
optoelectronic chaotic nodes
application I
chaotic communication
application II optimal network configurations
Group Synch and Chimeras
2
CO-AUTHORSHIP NETWORK
3
ref: Barabási et al., Physica A 311, 590 (2002)
DYNAMICS ON A COMPLEX NETWORK
Aij
j
i
N
dxi (t )
 Fi  xi (t )   i  Aij H j  x j (t   ij ) 
dt
j 1
4
 What is the dynamics of a single node ?
 How does the network structure Aij(t)
influence synchronization xi(t)?
 Do certain network configurations promote
group synch ?
 Chimera states?
5
 Study the dynamics of a network of nominally
identical time-delayed chaotic oscillators with
experiments & an accurate numerical model
6
optoelectronic chaos
laser
diode
optical
modulator
fiber
optic
cable
photodetector
data out
electronic
amplifier
data in
7
optoelectronic chaos
P0
P(t)
V(t)
 V

P  P0 cos 
 0 
 2V

2
V
P0
~10 cm
P0
P
P
(mW)
V
0
-6
-4
-2
0
V
2
4
6
8
optoelectronic chaos
V(t)

time-delayed feedback
9
ref: Y. Chembo Kouomou et al., Phys. Rev. Lett. 95, 203903 (2005).
optoelectronic chaos
simulation
P0
  1.15
x(t ) 
 V (t )
2V
laser power
  2.00
  2.45
  3.05

 RGP0
2V
feedback strength
experiment
  4.30
0
400 0
time (ns)
ref: A. B. Cohen et al., Phys. Rev. Lett. 101, 154102 (2008).
400
time (ns)
10
www.handsonresearch.org
January 2008
Institute for Plasma Research
Gandhinagar, India
July - August 2009
Universidade Federal do ABC
São Paulo, Brazil
August 2010
University of Buea
Buea, Cameroon
Next School : June 2012, Shanghai Jiao Tong University!
11
optoelectronic chaos
V(t)
bandpass
filter
time delay
du(t )
 Au(t )  B cos 2  x(t   )  0 
dt
x(t )  Cu(t )
12
optoelectronic chaos
V(t)
dac
dsp
adc
ram
digital signal
processing board
13
ref: T. E. Murphy et al., Phil. Trans. R. Soc. A 368, 343 (2010).
optoelectronic chaos
14
synchronization
-Christiaan Huygens (1665)
di
K N
 i   sin  j  i 
dt
N j 1
Kuramoto model (1975)
15
Communication with Synchronized
chaotic lasers
Demonstrations of laser sync
(rr and Thornburg PRL 1994
Sugawara et al., PRL 1994)
Proposal for optical chaotic comm
(Colet and rr Opt. Lett. 1994)
Demo of optcal chaotic comm
(Vanwiggeren and rr 1998,
Goedgebuer et al., 1998)
Hidden messages
Injecting a message into the
transmitter laser “folds” the data
into the chaotic fluctuations. The
receiver reverses this process,
thereby recovering a high-fidelity
copy of the message.
EDFA: erbium-doped fiber amplifier.
Lou Pecora, Physics World, 1998
17
Argyris et al, Nature 437, 343 (2005)
18
Transmitter and receiver systems
Optoelectronic systems
Claudio Mirasso
Leader of the EU consortium
Spotted here in Guanajuato,
Mexico (2005)
19
The History of Camouflage
The term camouflage comes from the French word
camoufler meaning "to blind or veil." Camouflage, also
called protective concealment, means to disguise an object,
in plain sight, in order to conceal it from something or
someone.
In the late 1800s, an American artist named Abbott Thayer made an
important observation about animals in nature that became a useful
tool in developing modern camouflage. After studying wildlife, Thayer
noticed that the coloring of many animals graduated from dark, on
their backs, to almost white on their bellies. This is an important
property that is very useful in modern camouflage. This gradation
from dark to light breaks up the surface of an object and makes it
harder to see the object as one thing. The object loses its threedimensional qualities and appears flat. This tendency to break up and
flatten the surface of an object also appears in the artistic movement,
Cubism, which was occurring during this same time period.
Camouflage, as we know it today, was born in 1915 when the French
army created a new unit called the camouflage division. Artists were
among the first people the French army called in to help develop
camouflage for use during WWI.
after Picasso's
Seated Woman, 1909
20
21
Frequency Hopping Spread Spectrum
1942 Patent: “Secret Communication System”
Hedy Kiesler Markey & George Antheil
22
Message communication at 1 Gbit/s
10 Gbits/s, Bit Error Rates ~ 10 -12 with error correction techniques
23
REPRESENTATIONS of NETWORKS of CHAOTIC OSCILLATORS
directed graph
j
adjacency matrix
Aij
 A11
A
21

A


 AN 1
i
A12
A22
A1N 




ANN 
coupled equations of motion
N
dxi (t )
 F  xi (t )    Aij H  x j (t   ) 
dt
j 1
24
CHAOTIC SYNCHRONIZATION
synced network
x1 (t )  x2 (t ) 
 x N (t )  s(t )
synced equations
N
dxi (t )
 F  xi (t )    Aij H  x j (t   ) 
dt
j 1
 k1  k2 
 kN
N
ki :  Aij
j 1
25
unidirectional coupling, N = 2
1 0 
A

1 0 
  1, 0
ref: A. Argyris et al., Nature 438, 343 (2005).
bidirectional coupling, N = 2
1
k
k
k = 0.2
k = 0.3
2
k 
1  k
A

k
1

k


  1,1  2k 
x1 (t )  x2 (t )
x1 (t )  x2 (t )
ref: T. E. Murphy et al., Phil. Trans. R. Soc. A 368, 343 (2010).
26
CHAOTIC SYNCHRONIZATION
synced network
x1 (t )  x2 (t ) 
 x N (t )  s(t )
synced equations: diffusive coupling


dxi (t )
 F  xi (t )    Aij H  x j (t   )   ki H  xi (t   )
dt
 j i

N
 F  xi (t )    Lij H  x j (t   ) 
j 1
N
ki :  Aij
j 1
27
application II
optimal networks
Q: can we determine how well a particular topology
of chaotic oscillators will synchronize?
SYNCHRONIZABILITY
 over what range of coupling strengths 
does the network exhibit synchrony?
 what is the coupling cost min for the network
to synchronize?
 how fast (emt) does the network converge to a
synchronous state?
28
application II
optimal networks
Q: can we determine how well a particular topology
of chaotic oscillators will synchronize?
A: yes, the variance of eigenvalues s of L is a
measure of these synchronization properties:
N
1
2
s  2
i  

d ( N  1) i 2
d :
2
m number of links

N number of nodes
networks that minimize s have optimal
synchronization properties.
ref: T. Nishikawa and A. E. Motter, Proc. Natl. Acad. Sci. U. S. A. 107, 10342 (2010).
29
application II
optimal networks
binary directional coupling, N = 4
a quantized number of links m = q (N  1)
minimizes variance s
q = 1, 2, … , N
30
application II
optimal networks
1
 mt

(
t
)

x
(
t
)

x
(
t
)
~
e
sync. error =

i
j
N ( N  1) i , j
e mt
synchronization floor
31
Four-node network:
32
application II
optimal networks
less can be more!
ref: B. Ravoori, et al., Phys. Rev. Lett. 107, 034192 (2011).
33
Influence of network structure:
Does the synchronizability measure s completely characterize the
synchronization properties?
For co-spectral configurations with m links, does the connection geometry
matter?
L(n )
0

 1

1

 1

0 0 0

1 0 0
0 1 0

0 0 1 
  {0,1,1,1}
diagonalizable (linearly independent
eigenvectors)
0 0 0

 1 1 0
L(s )  
0 1 1

 0 0 1

0

0
0

1 
  {0,1,1,1}
nondiagonalizable (linearly-dependent
eigenvectors)
34
Influence of network structure:
 nondiagonalizable configurations have an initial polynomial
convergence transient.
nondiagonalizable
 t  
1
 xi t   x j t .
N N  1 i , j
Initial polynomial transient!
diagonalizable
exponential convergence!
Experiment
Simulation
nondiagonalizable
diagonalizable
nondiagonalizable
diagonalizable
35
Influence of network structure:
 nondiagonalizable
configurations also exhibit greater variability among different
realizations and are in general more sensitive to perturbations. Hence we term
these configurations as sensitive.
nonsensitive configuration
sensitive configuration
Experiment
Simulation
sensitive
nonsensitive
sensitive
nonsensitive
36
B. Ravoori et al., Phys. Rev. Lett. 107 (034102 ), 2011.
Ring of coupled oscillators
37
38
Group Synch Experiments
Phys. Rev. E 76, 056114 (2007) Sorrentino and Ott
work in progress – Williams, Dahms, Schöll, Sorrentino
Nodes 1 and 3 : identical feedback strength
Nodes 2 and 4 : identical (but different) feedback strength
The two sets of nodes have different chaotic attractors
39
Group Synch Experiment
40
Chimeras
Kuramoto, Battoghtokh 2002
Nonlinear Phenom. Complex Syst. 5,
380 (2002).
Abrams, Strogatz PRL 2004
The Chimera is any mythical animal
with parts taken from various
animals and, more generally,
an impossible or foolish fantasy
(Wiki)
Sync and desync coexist in ensembles of non-locally coupled, identical nonlinear dynamical phase
oscillators
Can they be observed in real experimental systems?
Discussions with Y. Kuramoto, S. Strogatz, Carlo Laing, …………K. Showalter!
100 logistic maps
41
Nonlocally coupled chaotic logistic maps
(Also, coupled Rössler oscillators)
Omelchenko et al., PRL
42
Chaotic Iterated Map
φt
φt+1 = 0.85 [1-cos(φt)]
Time
• Can be implemented experimentally
43
43
Experimental Set Up
Bellerophon on Pegasus spears
the Chimera, 425–420 BC
44
These types of systems can be experimented on in
different world locations – in Beijing………
45
Spatial Light Modulator
Incoming light
Phase-shifted reflected light
Cover Glass
Grounded
Electrode
Liquid Crystal
Mirror
Pixel Electrodes
VSLI (chip)
• http://www.bnonlinear.com/
46
46
How to Make Maps
•
•
•
•
•
0) Partition the SLM screen into regions
1) Record intensity with camera
2) Couple regions with computer
3) Update phase shifts applied by SLM
4) Go to 1
Intensity measured by camera
φ0 φ1 φ2 φ3
φ4 φ5 φ6 φ7
φ8 φ9 φ10 φ11
φ12 φ13 φ14 φ15
Scaling (by computer)
Updated phase
Coupling between elements (In computer)
47
47
Coupled Iterated Maps
Local dynamics
R
Diffusive coupling
is coupled to
φ0 φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 φ10 φ11 φ12 φ13 φ14 φ15
Periodic Boundary Conditions
•Implemented experimentally!
48
48
1 D Parameter space
49
1d ring (per. bound. cond.)
(Chimera)
Simulation
Experiment
Three representative examples showing intensity as a function of
position in the 1d system.
At high coupling strengths, the patterns are coherent (A), and at low
coupling strengths they are incoherent(C).
Intermediate coupling strengths have coexisting domains of coherence
and incoherence (B). Chimera States
50
2d System
R
is coupled to
φ00 φ01 φ02 φ03 φ04 φ05 φ06 φ07
φ10 φ11 φ12 φ13 φ14 φ15 φ16 φ17
R
φ20 φ21 φ22 φ23 φ24 φ25 φ26 φ27
φ30 φ31 φ32 φ33 φ34 φ35 φ36 φ37
φ40 φ41 φ42 φ43 φ44 φ45 φ46 φ47
φ50 φ51 φ52 φ53 φ54 φ55 φ56 φ57
φ60 φ61 φ62 φ63 φ64 φ65 φ66 φ67
φ70 φ 71 φ 72 φ 73 φ 74 φ75 φ76 φ77
51
51
2D Parameter space
52
Compare 1D and 2D systems
53
53
HandsOn Research, Shanghai
Waiting to begin!
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Spiral Wave Chimeras?!
Excitation threshold, refractory period, time delayed feedback, noise
55
Spiral Wave Chimeras
S. Shima and Y. Kuramoto, Phys. Rev. E 69, 036213 (2004)
56
summary
Chaos, communication, chimeras
Optoelectronic time-delayed feedback loops are a good
testbed for studying nonlinear dynamics. Application
to chaotic communication.
Networks with optimal sync properties – network topology
influences convergence to synchronization.
Study synchronization in large 1d and 2d networks of
dynamical nodes – liquid crystal spatial light
modulators
Chimera states observed experimentally – work in
progress!
57
ACKNOWLEDGEMENTS
58
[0] A. Hagerstrom, T. E. Murphy, R. Roy, P. Hoevel, I. Omelchenko and E. Schoell
“Experimental Observation of Chimeras in Coupled Map Lattices,” accepted for
publication (2012).
[1] B. Ravoori, A.B. Cohen, J. Sun, A. E. Motter, T. E. Murphy and R. Roy,
“Robustness of Optimal Synchronization in a Real Network,”
Physical Review Letters 107, 034192, 1-4 (2011).
[2] A. B. Cohen, B. Ravoori, F. Sorrentino, T. E. Murphy, E. Ott and R. Roy, “Dynamic
synchronization of a time-evolving optical network of chaotic oscillators,”
Chaos 20, 043142, 1-9 (2010).
[3] T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. B. Schmitt, A. V. Setty, F. Sorrentino,
C. R. S. Williams, E. Ott and R. Roy, “Complex Dynamics and Synchronization of
Delayed-Feedback Nonlinear Oscillators,”
Philosophical Transactions of the Royal Society A 368, 343-366 (2010).
[4] B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott and R. Roy,
“Adaptive synchronization of coupled chaotic oscillators,”
Physical Review E 80, 056205, 1-4 (2009).
[5] A. B. Cohen, B. Ravoori, T. E. Murphy and and R. Roy,
“Using Synchronization for Prediction of High-Dimensional Chaotic Dynamics,”
Physical Review Letters 101, 154102, 1-4 (2008).
59
Those who understand others are intelligent
Those who understand themselves are enlightened
Those who overcome others have strength
Those who overcome themselves are powerful
Those who know contentment are wealthy
Those who proceed vigorously have willpower
Those who do not lose their base endure
Those who die but do not perish have longevity
Chapter 33, Tao Te Ching
60