1

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

2

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

3

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

4

1,000,000 Random Digits (RAND)

George W. Brown (RAND), 1949

5

Random Number Generators (RNGs)

• Computer algorithms: Pseudo-random number generators (PRNG)

• Classical, deterministic systems with large numbers of parameters:

6

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

7

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).

8

random numbers and public key cryptography

8

Feb 2012

9

Physical Random Number Generation

• Sir Francis Galton, “Dice for Statistical Experiments”, Nature 42, p. 13-14, 1890.

10

Quantum Mechanical RNG

• Photon-counting RNG(www idquantique.com)

• Geiger-counter RNG(built with Arduino!)

11

Testing for Randomness• How do you know that numbers are truly random?

12

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

13

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

14

Random Number Generation Today

• Currently used in all Intel “Ivy Bridge” processors• Operates at 3 Gb/s• Continuously re-seeds a PRNG

15

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).

16

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 !

17

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).

18

The World’s Fastest DiceNature Photonics Vol 2 (2008)

19

• “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

20

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)?

21

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)

22

Optical Noise Spectrum

• System generates broadband optical signal that can be spectrally separated into many bands

23

Electrical Noise Signal

• Very fast fluctuations (10 GHz bandwidth) limited by electrical speed and optical

bandwidth

• Macroscopic signal (~ 1 V)• Matches theoretical distribution

24

Statistical Test Results

• 109 bits tested (120 MB file)• Post-processing to suppress correlations• Passes all NIST Statistical tests

25

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:

26

dynamics on an interacting complex network of

optoelectronic chaotic nodes

I Group Synchrony

II Chimeras

III Symmetry and Sync

Outline

27

dynamics on an interacting complex network of

optoelectronic chaotic nodes

I Group Synchrony

II Chimeras

III Symmetry and Sync

28

DYNAMICS ON A COMPLEX NETWORK

i ijAj

[ ]1

( ( )( ) ) ij

N

i ii jji

jj

i Add

t ttt

ε τ=

+ −= ∑x xF Hx

29

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

30

Study the dynamics of a network of nominally identical time-delayed chaotic oscillators with experiments & an accurate numerical model

31

optoelectronic systems

laser diode

optical modulator

photo-detector

electronic amplifier

fiber optic cable

data in

data out

32

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

33

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

34

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

35

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

36

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

37

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

39

timeseries, 1881 photons/round trip

40

Distribution compared to Poisson,1881 photons/round trip

41

timeseries, 487 photons/round trip

42

Distribution compared to Poisson,487 photons/round trip

43

timeseries, 125 photons/round trip

44

Distribution compared to Poisson,125 photons/round trip

45

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

46

more d=2 embeddingsP0 Td = 3200 P0 Td = 800

P0 Td =200 P0 Td =12.5

47

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

)

48

Movie

𝛽𝛽 = 5.9

Plotting softwareMayavi & Python

49

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

50

Aaron counting photons

51

synchronization

“odd kind of sympathy”-Christiaan Huygens (1665)

( )1sini

N

ij

ijd Kdt N

θωθ θ=

= + −∑Kuramoto model (1975)

52

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

53

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

54

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

55

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).

11

Aκ κ

κ κ−

= −

{ }1,0Λ =

{ }1,1 2κΛ = −

56

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)

57

Ring of coupled oscillators

Caitlin Williams

58

59

“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).

60

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)

61

Experimental coupling scheme

62

Group Sync

63

Seven node simulation

64

65

Correlation Functions

66

Extension to other networks