Synchronization in complex network topologies Ljupco Kocarev Institute for Nonlinear Science, University of California, San Diego
Dec 14, 2015
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