Post on 01-Feb-2018
Bibliography
• Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence. Springer, Berlin.
• Pikovsky, A., Rosenblum, M. and Kurths, J. (2001). Synchronization. A universalconcept in non-linear sciences. Cambridge Nonlinear Science Series 12, University Press,Cambridge.
• Manrubia, S.C., Mikhailov, A.S. and Zanette, D.H. (2004). Emergence of DynamicalOrder. Synchronization PHenomena in Complex Systems. World Scientific, Singapore.
KURAMOTO’S MODEL ON NETWORKS
φi = ωi + k
N∑
j=1
Jij sin(φj − φi)
Jij = 1 if j acts on i. Otherwise, Jij = 0.
Jij: adjacency matrix → distance between oscillators
IDENTICAL OSCILLATORS:
φi =N
∑
j=1
Jij sin(φj − φi)
Full synchronization can be easily proven stable if:
• Jij = Jji (global; Lyapunov function)
• The network is regular,∑
j Jij = z (local)
An external oscillator (frequency Ω) coupled to oscillator 1 with strength a:
φi =N
∑
j=1
Jij sin(φj − φi) + aδi1 sin(Ωt− φi)
Eur. Phys. J. B 43, 97 (2005)
NUMERICAL RESULTS
• N = 1, 000 oscillators
• regular network, z = 2
• a = 10−3
• average phase: φ(t) = N−1∑
i φi(t)
• phase dispersion: σφi=
[⟨
(φi − φ)2⟩]1/2
0 3 6 9 12 15
10-7
10-6
10-5
10-4 Ω = 0.1 Ω = 2 Ω = 6 Ω = 20
σφ
d
-6 -4 -2 0 2 4 60
5x10-6
1x10-5
d=5 d=8 d=12 σφ
Ω
0.1 1 10
10-7
10-6
10-5
10-4
d=0 d=3 d=5 d=8 d=12
σφ
Ω
-0.8 -0.6 -0.4 -0.2
0.50
0.75
1.00
d > 6
d = 4
d = 5
d = 6
d = 3
d = 1
d = 2
d = 0
sin
φ
cos φ
0 3 6 9 12 15
10-3
10-2
10-1
100
a=0.001 a=1 a=3 a=10
σ φ /a
d
ANALYTICAL RESULTS:
• Linearization
φi(t) = φ(t) + aψi(t)
φ(t) = φ0 + aΦ(t)
• Expanding to the first order in a:
Φ =1
N
∑
ij
Jij(ψj − ψi) +1
Nexp(iΩt)
ψi = −Φ +∑
j
Jij(ψj − ψi) + δi1 exp(iΩt)
[sin → exp] (!)
• Propose ψi(t) = Ai exp(iΩt)
LA = b → A = L−1b
with A = (A1, A2, . . . , AN)
Lij = (z + iΩ)δij − Jij + 1N
∑
k Jkj
bi = δi1 −1N
• The amplitudes are complex: Ai = |Ai| exp(iϕi)
0 3 6 9 12 15
10-4
10-3
10-2
10-1 Ω = 0.1 Ω = 2 Ω = 6 Ω = 20
|A|
d
0 3 6 9 12 15
-5
-4
-3
-2
-1
0 Ω = 0.1 Ω = 1 Ω = 2 Ω = 6 Ω = 20
ϕ
d
APPROXIMATE EXPLICIT SOLUTION OF A = L−1b
Lij = (z + iΩ)δij − Jij + 1N
∑
k Jkj → L = (z + iΩ)I − J
with Jij = Jij −1N
∑
k Jkj
• Power expanding:
L−1 =[I − (z + iΩ)−1J ]−1
(z + iΩ)=
∞∑
m=0
J m
(z + iΩ)m+1
• Moreover
J(m)ij ≈ J
(m)ij −
zm
N
This gives
Ai =∞
∑
m=0
J(m)i1
(z + iΩ)m+1+
i
NΩ
• J(m)i1 : number of paths if length m from 1 to i.
• Small distances:
Ai ≈ (z2 + Ω2)−di+1
2 exp
[
−i(di + 1) tan−1 Ω
z
]
• Large distances:
Ai ≈i
Ω
−J0
(
1 +Ω2
z2
)
−di
2
exp
(
−idi tan−1 Ω
z
)
+1
N
• Large frequencies:
Ai ≈i
Ω
(
−δi1 +1
N
)
10-7
10-6
10-5
10-4
random I
σφ
0 3 6 9 12 15
10-7
10-6
10-5
10-4 random II
σφ
d
CHAOTIC (ROESSLER) OSCILLATORS
• Coupled oscillators
xi = −yi − zi + k∑
j Jij(xj − xi) + aδi1 sin Ωt
yi = xi + 0.2yi + k∑
j Jij(yj − yi)
zi = 0.2 + zi(xi − c) + k∑
j Jij(zj − zi).
• Measureσ
ri=
(⟨
|ri − r|2⟩)1/2
with
ri = (xi, yi, zi) and r = N−1∑
i ri
0 3 6 9 12 1510-6
10-5
10-4
10-3
Ω = 0.1 Ω = 0.5 Ω = 1 Ω = 3
σr
d
0.0 0.5 1.0 1.5 2.0 2.5 3.010-6
10-5
10-4
10-3
d = 0 d = 3 d = 7 d = 12
σr
Ω