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