phase synchronization of an ensemble of weakly coupled ...coupled oscillators: a paradigm of sensor...

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Faculty of Military Sciences

15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal

Phase synchronization of an ensemble of weakly coupled oscillators: A paradigm of sensor fusionPhase synchronization of an ensemble of weakly coupled oscillators: A paradigm of sensor fusion

Ariën J. van der Wal

Netherlands Defence Academy (NLDA) 15th ICCRTS

Santa Monica, CA, June 22-24, 2010

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ContentsContents

Introduction Synchronization as a paradigm for sensor data fusion

What is synchronization ? Phase Frequency

Examples of synchronization How do two systems synchronize: phase transition analogy Multi- oscillator synchronization in practice: Preliminary

results

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IntroductionIntroduction

Why are we interested in synchronization ? Motivation: try to understand mechanisms of sensor fusion Emergent behavior: What is it and when does it occur? Simple oscillators provide a good model for studying these We focus on phase synchronization as a basic mechanism for

inducing co-operative behavior Is it possible to extend the paradigm to real applications,

e.g. in modelling military sensor networks ?

Systems studied: 1. Non-linear coupling of 2 linear oscillators 2. Non-linear coupling between N linear oscillators3. Linear coupling of 2 non-linear oscillators

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Examples of synchronization processesExamples of synchronization processes

Biology: fireflies, yeast, algae, crickets Physiology: heart, brain, biological clocks, ovulation cycle Chemistry: chemical clocks Engineering: Power grids, distribution of time (UTC) Communication requires synchronisation at all OSI layers Physics: coherence of lasers and masers, phase transitions,

ferromagnetism, superconductivity, spin waves

SHOW physics demo: 3 metronomes synchronization presentatie\Synchronization of Three Metronomes.MP4

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SynchronizationSynchronization

History: Christiaan Huygens (Feb 1665) “Sympathie des horloges” 2 pendulum clocks suspended from the same beam will in a

relatively short period assume the same rythm if they are initially out-of-phase; they will eventually synchronize and lockin antiphase !

Constant phase difference between 2 oscillations:

Only possible if both have the same frequency:

Amplitude of oscillator can be chaotic

0 constant

1 20 f f

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The definition of phaseThe definition of phase

How can “phase” be defined for an arbitrary, periodic signal?

Different ways to define momentary phase: For a simple sine fœ[0,2p] For a periodic function phase plane; Poincaré map For a complex oscillator Hilbert transform

( )

( )

1 ( )( )

( ) ( ) ( ) ( )( )tan( ( ))( )

i t

x x t x

xy t P dt

z t x t iy t r t e zy ttx t

H x

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Two coupled linear oscillators (1)Two coupled linear oscillators (1)

2 oscillators, each with its own eigenfrequency :

with nonlinear interaction dependent on the phase differenceso that

with

1 1 12 1 2

2 2 21 2 1

( )

( )

d Udtd Udt

( )d udt

1 2

1 2

12 21( ) ( ) ( )u U U

1 2 12 ( )U

1,2

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Two coupled linear oscillators (2) Two coupled linear oscillators (2)

If synchronization occurs, we have:

So if real roots for this algebraic equation exist, we have found synchronous solutions !

A an example we take and find the graphical solutions of

is stable and unstable

sin 0ddt

0 ( )d udt

( ) sinu

-Δω

quqs

u

q

us

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Synchronization occurs when

This algebraic equation only has real roots iff the difference in eigenfrequencies lies within the interval of values of the function :

In our example we have:

“Arnold tongue”

( ) sinu

0 ( )d udt

( )u

1 2 synchronized

ω1 ω2

ε

0

min maxu u

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The common synchronization frequency of the two coupled oscillators follows from:

where is the phase difference from the stable graphical solution

Outside the entrainment region the motions are notsynchronous, but they can still influence each other significantly. (-> phase slips)As an example take

so that

with solution

1 12 0 2 21 0( ) ( )U U

sinddt

0

112 21 2( ) ( ) sinU U

22 21

22( ) 2arctan 1 tan( )t t

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Time dependence of phase difference outside the region of synchronization.

1 2 1 1.05

time

θ

2π phase slip

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9 fe

b10

Air coordination centre

Tactical Sensor Sensors

Air effectors

NEC in inhomogeneous networks NEC in inhomogeneous networks

UAV Ground Station

Remote Calculation

Station

Remote HQ

Image interpretationsystem

Coordination &Fusion centre

HQ


Modeling a homogeneous sensor networkModeling a homogeneous sensor network

Homogeneous network of N nodes (“agents”) acts as a distributed sensor (or detector)

Homogeneous networks are part of typical NEC networks Node composition: (analog) sensor, memory, decision taking Mathematical modelling: Node = oscillator Observable determines the oscillator frequency: Node =

parametric oscillator (or VCO ?) Contact between nodes through non-linear coupling K Study the dynamic behavior of the ensemble of N coupled

oscillators


Why is synchronization important ?Why is synchronization important ?

Sensor network: each member of the population is represented by a phase oscillator

Synchronization on physical layer, not on protocol layer: faster and more accurate

Greater robustness, fault tolerance, scalability, smallcomplexity self-synchronization

Ultimate goal: local information storage, propagation of information, distributed, “soft” decision taking

Redistribution of mobile sensors to more effectively sample the environment in presence of measurement noise

Propagation and fusion of analog information without a central fusion master


N linear oscillators N linear oscillators

Kuramoto model ensemble of N nearly identical oscillators symmetric distribution of eigenfrequences global coupling strength evolution of oscillator phase given by

Stationary synchronization (mean-field approximation)

complex order parameter r:

1sin( ) ( 1,..., )

Nk

k j kj

d t K t t k Ndt N

1

1k

Nii

kre e

N

sin( ) ( 1,..., )kk k

d tKr k N

dt

0K ( ) ( )g g


Sensors acting as detectorsSensors acting as detectors

Distributed, dense sensor network Detection as a stochastic process:

if an event is detectedif no event is detected

Probability of detection p0 If the network is sufficiently large the phase rate

converges to :

0i 1i

** 1

0 1 0 0

1

(1 )N

k kkN

kk

cd p pdt c

*0

0 0

*1

1 0

arcsin 1(0)

arcsinj

with probability pKr

with probability pKr

**( ) (0)j jt t


Synchronisation phase diagram N=400 oscillatorsSynchronisation phase diagram N=400 oscillators

0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 = 0.5 400 oscillators

Syn

chro

nize

d fr

actio

n

(1-2/)

SYNCHRONIZED

UNSYNCHRONIZED

interaction strength K

sync

hron

ized

fract

ion


Integration time step = 0.01

How fast is synchronization for N=400 ?How fast is synchronization for N=400 ?

0 10 20 30 40 50 60 70 80 90 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9K=1.700000000000000

time t

orde

red

frac

tion

orde

red

fract

ion


Results of simulations N=400Results of simulations N=400

presentatie\freq N=400.avi presentatie\phase N=400.avi

frequency phase

0 50 100 150 200 250 300 350 400-5

-4

-3

-2

-1

0

1

2

3

4

5K=5.000000000000000E-001

oscillator #

devi

atio

n fr

om c

entr

al f

requ

ency

0 50 100 150 200 250 300 350 400-8

-6

-4

-2

0

2

4

6

8K=5.000000000000000E-001

oscillator #

osci

llato

r ph

ase

mod

2


SynergySynergy

The basic notion of synergy is:

or at least:

Non-lineartity is an essential ingredient for understanding sensor data fusion !

Classical approach via Bayesian networks, DS theory and/or fuzzy (belief and plausability) measures

In the present study we focus on a different approach: the paradigm of phase transitions in physics

( ) max( ( ), ( ))g A B g A g B

( ) ( ) ( )g A B g A g B

1 ( ) ( ) ( ) ( ) ( )g A B g A g B g A g B


ConclusionsConclusions

Nonlinear coupling of 2 linear oscillators results in very fastphase synchronization, provided that the interaction is strongenough. Synchronization of 2 nonlinear oscillators occurs already at veryweak coupling. In a non-linear globally interacting many-particle system we observe spontaneous (partial) synchronization above a critical interaction strength.

The fast and spontaneous synchronization of globally interactingsystems is a form of emergent behavior and may be exploited as a mechanism for military smart sensor networks.

phase synchronization of an ensemble of weakly coupled ...coupled oscillators: a paradigm of sensor...

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