Experiments on Real Networks CHAOS and... · example of partnership between quantum fluctuations...
Transcript of Experiments on Real Networks CHAOS and... · example of partnership between quantum fluctuations...
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Rajarshi RoyInstitute for Physical Science and Technology
Department of PhysicsInstitute for Research in Electronics and Applied Physics
University of Maryland, College Park
Hands On School in Nonlinear DynamicsIPR, Gandhinagar
February, 2015
Noise, Chaos and SynchronyExperiments on Real Networks
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What is Randomness?(what are random numbers?)
• Arbitrary, unpredictable• Statistical properties, uniform distribution Given Lack of bias: Uncorrelated:
• Is noise random? In theory, can be infinite-bandwidth white noise, but
in reality, it always has a finite bandwidth Small amplitude, potentially biased, limited speed
xi ∈ [0,1]
P[xi =1] = 12
P[xi =1 | xi−1,xi−2,...] = 12
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Uses of Random Numbers
• Gaming: lotteries, slot machines, video poker, roulette
• Monte Carlo simulation mathematically modeling noisy systems Establishing random initial conditions
• Encryption and security commerce (https) secure communication financial transactions
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1,000,000 Random Digits (RAND)
George W. Brown (RAND), 1949
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Random Number Generators (RNGs)
• Computer algorithms: Pseudo-random number generators (PRNG)
• Classical, deterministic systems with large numbers of parameters:
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Pseudorandom Number Generators
• Generate a sequence of bits, starting from a finite-length “seed”
• Rely on “one-way functions”: arithmetic operations or algorithms that are difficult
to reverse• If the seed and algorithm is known, the future (and
sometimes past) sequence can be reproduced exactly.• All PRNG sequences eventually repeat Ex: Mersenne Twister: 219937 − 1
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Psuedorandom Number Generators
• “Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin.”
– John von Neumann, 1951
• Pseudorandom number generatorscan be designed to pass the tests for randomness, but are still vulnerable to hacking, if the initial state (seed) and the process is known
• In 2007, a group found Microsoft’s pseudo-RNG to be vulnerable to backward attacks
Dorendorf, L., Gutterman, Z. &Pinkas, B., “Cryptanalysis of the random number generator of the Windows Operating System,” in Proc. 14th ACM Conf. Computer Commun. Security 476-485 (2007).
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random numbers and public key cryptography
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Feb 2012
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Physical Random Number Generation
• Sir Francis Galton, “Dice for Statistical Experiments”, Nature 42, p. 13-14, 1890.
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Quantum Mechanical RNG
• Photon-counting RNG(www idquantique.com)
• Geiger-counter RNG(built with Arduino!)
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Testing for Randomness• How do you know that numbers are truly random?
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Evaluating Random Sequences
• NIST publishes a comprehensive battery of statistical tests – look for patterns in bit sequences Evaluate complexity and statistical distributions Uses a sequence of finite length (1000-1Mbit sequences) Caution: Many pseudo-random numbers pass the tests!
• Entropy considerations – more fundamental measure of unpredictability from system model
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How Fast Can You Generate Random Numbers?
• Pseudorandom number generation: Limited by CPU clock “Cryptographically secure” RNGs often use large-
integer arithmetic that is computationally intensive Dedicated hardware PRNGs are available
• Physical Random Number Generators: Photon counting: up to 40 Mb/s Electrical noise: up to Gb/s
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Random Number Generation Today
• Currently used in all Intel “Ivy Bridge” processors• Operates at 3 Gb/s• Continuously re-seeds a PRNG
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Random numbers from chaotic lasers
• Two chaotic lasers with optical feedback used to generate two signals converted to bit strings
A. Uchida, et. al, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photonics 2, 728-732 (2008).
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Random numbers from chaotic lasers
• Produced bit sequences that pass the NIST and Diehard tests
• Speed limited by chaos bandwidth
• Speed: 1.7 Gb/s !
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Random numbers from chaotic lasers
• Resultant bit sequence:
A. Uchida, et. al, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photonics 2, 728-732 (2008).
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The World’s Fastest DiceNature Photonics Vol 2 (2008)
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• “The present experiment can thus be regarded as a beautiful example of partnership between quantum fluctuations (spontaneous emission) and chaotic dynamics at the macroscopic level. It illustrates how large, easily detected fluctuations can emerge from truly random quantum fluctuations, connected by the bridge of a nonlinear dynamical system — the semiconductor laser with optical feedback. The three elements of quantum fluctuations, nonlinearity and time-delayed feedback work together in this experiment to produce a virtually inexhaustible store of random numbers at the highest speeds achieved so far. This is an example of nature helping us with a difficult (according to Von Neumann, impossible) mathematical task.”
From quantum fluctuations to chaos
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Spontaneous EmissionCaitlin Williams, Adam Cohen, Julia Salevan, Xiaowen Li, Thomas Murphy
• Occurs in all optical lasers and amplifiers• Quantum mechanical in origin• Produces large, easily observed fluctuations in optical
intensity
CAN WE USE SPONTANEOUS EMISSION FOR GENERATING RANDOM NUMBERS (FAST)?
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Experimental Setup
• Use commercially-available fiber optic components usually found in telecom systems
• Amplified spontaneous emission is unpolarized – two independent noise sources
• Balanced detection helps to reduce bias
• Generation rate: 12.5 Gb/sC. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, "Fast physical random number generator using amplified spontaneous emission" Opt. Express 18, 23584-23597 (2010)
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Optical Noise Spectrum
• System generates broadband optical signal that can be spectrally separated into many bands
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Electrical Noise Signal
• Very fast fluctuations (10 GHz bandwidth) limited by electrical speed and optical
bandwidth
• Macroscopic signal (~ 1 V)• Matches theoretical distribution
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Statistical Test Results
• 109 bits tested (120 MB file)• Post-processing to suppress correlations• Passes all NIST Statistical tests
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We’ve come a long way
• 1949: Approximately 1 random digit per second (3.3 bits/sec)It took 11.6 days to generate the numbers in this book!
THEN:
• 2014: 20,000,000,000 bits/secIt takes only 166 µs to generate amillion random digits
NOW:
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dynamics on an interacting complex network of
optoelectronic chaotic nodes
I Group Synchrony
II Chimeras
III Symmetry and Sync
Outline
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dynamics on an interacting complex network of
optoelectronic chaotic nodes
I Group Synchrony
II Chimeras
III Symmetry and Sync
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DYNAMICS ON A COMPLEX NETWORK
i ijAj
[ ]1
( ( )( ) ) ij
N
i ii jji
jj
i Add
t ttt
ε τ=
+ −= ∑x xF Hx
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What is the dynamics of a single node ?
How does the network structure Aij(t)influence synchronization xi(t)?
Do certain network configurations promote optimal synchronization ?
Group Synchronization
Chimera States
Symmetry and Sync
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Study the dynamics of a network of nominally identical time-delayed chaotic oscillators with experiments & an accurate numerical model
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optoelectronic systems
laser diode
optical modulator
photo-detector
electronic amplifier
fiber optic cable
data in
data out
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optoelectronic systems
V(t)
P0 P(t)
200 cos
2VP VP
π
π φ
= +
~10 cm
V
P0 P
-6 -4 -2 0 2 4 6
P
V
P0
0
(mW)
Vπ
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optoelectronic chaos
V(t)
time-delayed feedback
τY. Chembo Kouomou, Pere Colet, Laurent Larger and N. Gastaud, Phys. Rev. Lett. 95, 203903 (2005).
Chaotic Breathers in Delayed Electro-Optical Systems
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optoelectronic chaos
((2
)) VxV
ttπ
π=
feed
back
stre
ngth
experiment simulation
ref: A. B. Cohen et al., Phys. Rev. Lett. 101, 154102 (2008).
0
2PRG
Vπ
β π=1.15β =
2.00β =
2.45β =
3.05β =
4.30β =
time (ns) time (ns)0 400 0 400
lase
r pow
er
0P
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www.handsonresearch.org
January 2008Institute for Plasma Research
Gandhinagar, India
July - August 2009Universidade Federal do ABC
São Paulo, Brazil
August 2010University of BueaBuea, Cameroon
School : June 2012, Shanghai Jiao Tong University
2013, 2014, 2015 in Trieste, Italy, at Abdus Salam International Centre for Theoretical Physics
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optoelectronic chaos
[ ]20
( ) ( ) c )
( ) (
(os
)
xd t td
t
t
xtt
τ φβ= − +
=
−u
C
u
u
A B
V(t)
bandpass filter
time delay
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optoelectronic chaos
ref: T. E. Murphy et al., Phil. Trans. R. Soc. A 368, 343 (2010).
V(t)
digital signal processing board
adcdacdsp
ram
optoelectronic chaos
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timeseries, 1881 photons/round trip
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Distribution compared to Poisson,1881 photons/round trip
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timeseries, 487 photons/round trip
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Distribution compared to Poisson,487 photons/round trip
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timeseries, 125 photons/round trip
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Distribution compared to Poisson,125 photons/round trip
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time delay embeddings
𝑁𝑁1,𝑁𝑁2, 𝑁𝑁3, … ,𝑁𝑁𝑖𝑖 , … ,𝑁𝑁𝐿𝐿List of scalar measurements:List of vectors (d=2):
𝑁𝑁1𝑁𝑁1+𝜏𝜏𝑒𝑒
,𝑁𝑁2
𝑁𝑁2+𝜏𝜏𝑒𝑒,
𝑁𝑁3𝑁𝑁2+𝜏𝜏𝑒𝑒
, … ,𝑁𝑁𝑖𝑖
𝑁𝑁𝑖𝑖+𝜏𝜏𝑒𝑒, … ,
𝑁𝑁𝐿𝐿−𝜏𝜏𝑒𝑒𝑁𝑁𝐿𝐿
𝜏𝜏𝑒𝑒 =𝑇𝑇𝑑𝑑4
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more d=2 embeddingsP0 Td = 3200 P0 Td = 800
P0 Td =200 P0 Td =12.5
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Poincaré surfaces
λTD = 3200
λTD = 12.5pr
obab
ility
λTD = 200
model
𝑁𝑁𝑤𝑤(𝑡𝑡) 𝑁𝑁𝑤𝑤(𝑡𝑡)
𝑁𝑁𝑤𝑤(𝑡𝑡) 𝐼𝐼𝑤𝑤(𝑡𝑡)
𝑁𝑁 𝑤𝑤(𝑡𝑡−𝜏𝜏 𝑒𝑒
)
𝑁𝑁 𝑤𝑤(𝑡𝑡−𝜏𝜏 𝑒𝑒
)
𝑁𝑁 𝑤𝑤(𝑡𝑡−𝜏𝜏 𝑒𝑒
)
𝐼𝐼 𝑤𝑤(𝑡𝑡−𝜏𝜏 𝑒𝑒
)
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Movie
𝛽𝛽 = 5.9
Plotting softwareMayavi & Python
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Experimental Implementation
• Filter and delay: FPGAtwo-pole lowpass, 150 MHz clock
• Laser: 852 nm DFB• Photon counter: Si APD• EO modulator:
850 nm LiNbO3 Mach-Zehnder
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Aaron counting photons
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synchronization
“odd kind of sympathy”-Christiaan Huygens (1665)
( )1sini
N
ij
ijd Kdt N
θωθ θ=
= + −∑Kuramoto model (1975)
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Message communication at 1 Gbit/s
~ 10 Gbits/s, Bit Error Rates ~ 10-12 with error correction techniques
Professor Uchida’s book covers Random Number generation by chaotic lasers and many other topics !!
Atsushi Uchida
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11 12 1
21 22
1
N
N NN
A A AA A
A A
=
A
[ ]1
( ( )( ) ) iji
ji
N
j
tddt
tAt τε=
= + −∑ xx F x H
directed graph
adjacency matrix
coupled equations of motion
ji
ijA
REPRESENTATIONS of NETWORKS of CHAOTIC OSCILLATORS
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21 ( )) ) )( ( (Ntt tt= = = =x sx x
1:i ij
N
jk A
=
= ∑
1 2 Nk k k∴ = = =
CHAOTIC SYNCHRONIZATION
synced network
[ ]1
( ( )( ) ) iji
ji
N
j
tddt
tAt τε=
= + −∑ xx F x H
synced equations
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1 01 0
A =
unidirectional coupling, N = 2
ref: A. Argyris et al., Nature 438, 343 (2005).
1 2κ
κ
21( () )x xt t=21( () )x xt t≠
κ = 0.2 κ = 0.3
bidirectional coupling, N = 2
ref: T. E. Murphy et al., Phil. Trans. R. Soc. A 368, 343 (2010).
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Aκ κ
κ κ−
= −
{ }1,0Λ =
{ }1,1 2κΛ = −
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21 ( )) ) )( ( (Ntt tt= = = =x sx x
CHAOTIC SYNCHRONIZATION
synced network
synced equations: diffusive coupling
[ ] [ ]
[ ]1
( ) ( ) ( )
( )
( )
( )
ii ijij i
ii
j i
jj
j
N
A kt t
tL
tdt
t
t
d τε
τ
τ
ε
≠
=
= + −
−
−
−
= −
∑
∑
F H H
F
x x
x
xx
xH1
:i ij
N
jk A
=
= ∑
Master Stability function Pecora and Carroll, Phys. Rev. Lett. 80, 2109–2112 (1998)
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Ring of coupled oscillators
Caitlin Williams
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“Group synchrony”
• In group synchronization the local dynamics in synchronized clusters can be different from the dynamics in the other cluster(s).
• Extends the possibility of synchronization behavior to networks formed of heterogeneous dynamical systems.
• These synchronous patterns can be observed even when there is no intra-group coupling.
• Sorrentino and Ott have generalized the MSF approach to group synchronization, and recent work by Dahms et al. considers time-delayed coupling of an arbitrary number of groups.
F. Sorrentino and E. Ott, Phys. Rev. E 76, 056114 (2007)T. Dahms, J. Lehnert, and E. Schöll, Phys. Rev. E 86, 016202 (2012).
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Example of Group Sync – 4 nodesExperiments: Caitlin R. S. Williams, Thomas E. Murphy, rr
Theory: Francesco Sorrentino, Thomas Dahms, and Eckehard SchöllPRL 110, 064104 (2013)
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Experimental coupling scheme
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Group Sync
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Seven node simulation
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Correlation Functions
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Extension to other networks