synchronization in complex network topologies ljupco kocarev institute for nonlinear science,...

Synchronization in complex network topologies

Ljupco Kocarev

Institute for Nonlinear Science, University of California, San Diego

Outlook

• Chaotic oscillations and types of synchrony observed between chaotic oscillators

• Experimental and theoretical analysis of chaos synchronization; Stability of the synchronization manifold

• Synchronization in networks

Periodic and Chaotic Oscillations

Power Spectrum

Waveform x(t)

Power Spectrum

Waveform x(t)

),(xFx

dt

d

3 , nnx

Chaotic Attractor

)()( 0 tt(t) xxη

,))(( 0

tdt

dxDF )(0 tx

)0(d)(t

id

Lyapunov exponents:

) 0(

) (log

t

1 lim )) ( ( 0

d

t i dt it

x

• Phase Synchronization.

• Synchronization of switching.

• Others

Types of chaos synchronization

Complete Synchronization Partial Synchronization

• Identical synchronous chaotic oscillations.

• Generalized synchronized chaos.

• Threshold synchronization of chaotic pulses.

0)()(lim

tytxt

0)()(lim

tytxt

time

nT1nT )( 1 nn TFT

nt1n

t2nt

)(tx

)( )( ,)( )( yn

ttyxn

ttx

11,A22 ,A

const 21

)()( yn

txn

t

t

)(tx )(ty

Synchronization of chaos in electrical circuits.

3.0

-3.0

-2.5

-2.0-1.5

-1.0

-0.50.00.5

1.01.5

2.02.5

2.1-2.1 -1.5-1.0-0.5 0.0 0.5 1.0 1.5

PHASE PORTRAIT

)(1 tx

)(3 tx

Unidirectional coupling

N

R

C’C

rL

)f( 1x

)(1 tx)(3 tx

2~)( xtI

N

R

C’C

rL

)f( 1y

)(1 ty)(3 ty

2~)( ytI

)(1 tx

Driving Oscillator Response OscillatorCoupling

CI

)()(1

11 tytxR

IC

C

CR

-2 -1 0 1 2

-0.5

0.0

0.5)(f x

x

Synchronization Manifold

23132313

32123212

112121

])f([])f([

)(

yyyyxxxx

yyyyxxxx

yxgyyxx

The model:

C

L

Rcg

1The coupling parameter:

There exits a 3-dimensional invariant manifold:

33

22

11

yx

yx

yx

Synchronization of chaos: Experiment

1

devic e (1)

0,80

c hannels (0)

5000

buffer size

(10000)

40000.00

sc an rate

(4000 scans/ sec)

5000

# scans to read

at a time (1000)

3.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0-3.0 -2.0 -1.0 0.0 1.0 2.0

PHASE PORTRAIT

Hanning

window

1.0E+0

1.0E -11

1.0E -9

1.0E -7

1.0E -5

1.0E -3

40000 500 1000 1500 2000 2500 3000 3500

POWER SPECTRUM

2.5

-2.5

-2.0

-1.0

0.0

1.0

2.0

400 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

WAVEFORM X(t)

Hz

mSec

3.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0-3.0 -2.0 -1.0 0.0 1.0 2.0

PHASE PORTRAIT3.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0-3.0 -2.0 -1.0 0.0 1.0 2.0

PHASE PORTRAIT

3.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0-3.0 -2.0 -1.0 0.0 1.0 2.0

PHASE PORTRAIT3.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0-3.0 -2.0 -1.0 0.0 1.0 2.0

PHASE PORTRAIT

3.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0-3.0 -2.0 -1.0 0.0 1.0 2.0

PHASE PORTRAIT3.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0-3.0 -2.0 -1.0 0.0 1.0 2.0

PHASE PORTRAIT

Driving OscillatorDriving Oscillator Response OscillatorResponse Oscillator

UncoupledOscillators

Coupling belowthe threshold of synchronization

Coupling abovethe threshold of synchronization

Stability of the Synchronization Manifold:Identical Synchronization

0)0( , , )()(

),(

GyyxGyFy

,xxFxn

n

yx

)()()( ttt xy

j

iij x

FtDFtt

)(xDGxDF )(,)]0())(([)(

Synchronization Manifold:

Perturbations transversal to the Synchronization Manifold:

Linearized Equations for the transversal perturbations:

Driving System:

Response System:

x

y )(t

)(tx

0

0

Chaos Synchronization Regime

)]0())(([)(

)(

DGxDF

xFx

ttConsider dynamics in the

phase space ),( x

x

x

x

x

The parameter space

p1

p2

Synch

No Synch

No Synchronization Synchronization

A regime of dynamical behavior should have a qualitative feature that is an invariant for this regime.

- Projection of chaotic limiting set Transienttrajectories- Limit cycles

Synchronization of Chaos in Numerical Simulations

)(1 tx

)(3 tx

)(1 tx )(1 tx

)(1 ty )(1 ty

Simulation without noise and parameter

mismatch

Simulation with 0.4% of parameter mismatch

Attractor in the DrivingCircuit )1.0( max

Transversal Lyapunov exponent evaluatedfor the chaotic trajectory x(t) equals 03.0max Coupling: g=1.1

N

kk ik i ix D x f x

1

) (

i x

ik D

m - dimensional vector

- real matrixm m

N i,..., 1

H g Dik ikH

- real matrixm m

Assumptions:

ik g

- real number

Network with N nodes

Synchronization manifold:

Connectivity matrix:

) (ik g G N N

- real matrix

Nx x x ... 2 1

0 jij g

k k kH J ) (

k

- eigenvalue of the connectivity matrix

N k,..., 1

Variation equation:

) (ik g G

0 1

) (i ix f x

k kH i J ] ) ( [

}0 ) , ( : ) , {(max

Properties of the master stability function

}0 ) , ( : ) , {(max

• Empty set• Ellipsoid • Half plane

The master stability function for x coupling in the Rossler circuit.

The dashed lines show contours in theunstable region.

The solid lines are contours in the stable region.

) , ( max versus

Stable region:

) , (2 1 max

) , ( max

0 ...1 2 1 N N2

1

2

N) , (2 1 max

) , ( max 2

L G

Laplacian matrix

BN

2

A 2

Class-A oscillators

Class-B oscillators

B>1

Consider N nodes (dots); Take every pair (i,j) of nodes and connect them with an edge with probability p

),(, EVG pN

Erdős-Rényi Random Graph(also called the binomial random graph)

Power-law networks

Power-law distribution

=<k>

•Power-law graphs with prescribed degree sequence (configuration model, 1978)•Evolution models (BA model, 1999; Cooper and Frieze model, 2001)•Power-law models with given expected degree sequence (Chung and Lu, 2001)

Hybrid Graphs

Hybrid graph is a union of global graph (consisting of “long edges” providing small distances) and a local graph (consisting of “short edges” representing local connections). The edge set of of the hybrid graph is a disjoint union of the edge set of the global graph G and that of the local graph L.

G: classical random model power-law model

L: grid graph

Theorem 1. Let G(N,p) be a random graph on N vertices. For sufficiently large N, the class-A network G(N,q) almost surely synchronize for arbitrary small coupling. For sufficiently large N, almost every class-B network G(N,p) with B>1 is synchronizable.

Theorem 2. Let M(N, , d, m) be a random power-law graph on N vertices. For sufficiently large N, the class-A network M(N, , d, m) almost surely synchronize for arbitrary small coupling. For sufficiently large N, almost every class-B network M(N, , d, m) is synchronizable only if B

d

mN lim

2

power of the power-law

d expected average degree

m expected maximum degree

75 . 16592

N

0024 . 0 2

Consider a hybrid graph for which L is a circle with N=128.

Consider class-A oscillators for which A=1 and

10

Consider class-B oscillators for which B=40

pNG number of global edges a)

0

20

40

60

80

100

120

140

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p

γ2

b)

0

100

200

300

400

500

0.001 0.01 0.1 1

p

γN/γ2

p=0.005 p=0.01

Local networks Oscillators do not synchronize

Hybrid networks

Random networks

Power-law Oscillators may or may not synchronize

Binomial Oscillators synchronize

Power-law Oscillators synchronize

Binomial Oscillators synchronize

Conclusions

• Two oscillators may have different synchronous behavior

• Synchronization of identical chaotic oscillations are found in the oscillators of various nature (including biological neurons)

• Global edges improve synchronization