管曙光-复杂网络上的广义同步

• The distance of global GS
To characterize the occurence of global GS on the network, we define l g (ε ) = d (ε , i ) , where the ⋅ denotes the average on all nodes.
• The distance of global CS
Moreover, the global CS on the network can be characterized by the distance of global CS as 1 t1 lc (ε ) = ∫t1 X i − X i dt . t2 − t1
[5] X. F. Wang and G. Chen, Int. J. of Bifur. Chaos 12, 187 (2002). [6] Alex Arenas, et al., Phys. Rep. 469, 93 (2008). [7] 汪小帆，李翔，陈关荣，《复杂网络—理论与应用》，清华大学出版社，2006。 [8] 何大韧，刘宗华，汪秉宏，《复杂系统与复杂网络》，高等教育出版社，2009。
Yd
0
−30 −30
0
30
Yr
GS
• Synchronization on networks Recently, synchronization has been extended to the area on complex networks, such as small-world, scale-free, modular (community) networks, weighted networks, and gradient networks, etc.
id
400
5
: ranked by decreasing degree
500
0
0.5
1.0
1.5
2.0
2.5
3.0
ε
Color-map of matrix d (ε , i ) , showing the development of GS is from the hubs to the rest nodes in SF network
[1] L. M. Pecora et al., PRL 64, 821(1990). [2] N. F. Rulkov, et al., PRE 51, 980 (1995). [3] M. G. Rosenblum, et al., PRL 76, 1804 (1996). [4] S. Boccaletti, et al., Phys. Rep. 366, 1 (2002).
60 80 100 1.0 2.0 3.0
3 The development of GS on networks
(1) 500 non-identical Lorenz oscillators on SFN
1 25 100
r randomly in [28, 38] Scale-free network (BA)
20
200
15
i
d
300
10
∑a H (X
ij j
i
− X j ), i = 1,2,...N
(a)
25 20 15 10 5
SF network
150 200 250 300 0 2 4 6 8 10
(a) 300 identical Lorenz oscillators
id
(b)
1 50 100 0.4 0.3 0.2 0.1 0 0.2 0.4 0.6 0.8 1 0
• The auxiliary method
Response R Drive D Response R’
R and R’ have the same dynamics, but from different initial conditions R=R’ (CS) implies R=f (D) (GS)
The entrainment for nodes 4,104, and 250
(c) ε = 0.77 50% entrained (d ) ε = 3 all entrained
• Topology vs local dynamics
SF network
For heterogeneous networks, the hub nodes provide skeleton around which GS is developed.
Set ∆X i = X i − X i' , we can have & ∆X i ≈ [ DFi(X i ) − εki H ] ⋅ ∆X i , i.e., the entrainment of node i
id : randed by decreasing degree ir : randed by decreasing ri (LLE)
T
The local dynamics can be chosen as Lorenz equation : & xi = 10( yi − xi ) & yi = ri xi − yi − xi zi & zi = xi yi − 8 zi / 3, with ri uniformly distributed on [28, 28 + ∆r ].
150 200 250 300
(b) 300 identical Logistic maps
idBiblioteka ε• Networked hybrid oscillators
SF network
25 50 100 15 20
d
150 10 200 250 300 0 1 2 3 4 5 0
i
ε
Hybrid system: Lorenz oscillators: 95% Rossler oscillators: 5%
1 The background
• Chaos synchronization
In the past two decades, chaotic synchronization has been extensively studied, such as complete synchronization (CS), generalized synchronization (GS), phase synchronization (PS), and many others.
depends on the effective coupling strength εki .
SW network
Modular network
For homogeneous networks, the local dynamics (LLE) determines the development of GS.
(2) 300 identical Lorenz oscillators on SFN
1 50 20 100 15 25
150 10 200 250 300 5
r = 28 for all nodes
id
0
0.5
1.0
1.5
2.0
0
ε
Typical path: STC GS CS
(a) ε = 0 not entrained (b) ε = 0.68 80% entrained
[9] H. D. I. Abarbanel et al., PRE 53, 4528 (1996).
• The auxiliary method on networks
Node i
Auxiliary node i’
Oscillator i & X i = Fi ( X i ) − ε ∑ aij H ( X i − X j )
• The distance of local GS
To characterize the occurence of GS on networks, we define the following quantity 1 t2 ' d (ε , i ) = ∫t1 X i − X i dt . t2 − t1 Given (ε , i ), d (ε , i ) measures the distance between oscillator i and it auxiliary oscillator i '. If d (ε , i ) = 0, CS is achieved i and i' , implying node i has been entrained in the network.
• Motivation
So far, most works deal with CS and PS on complex networks. In real world, dynamics of oscillators on networks could be very complicated, and generally are different rather than identical. (1) Can GS occur on complex networks? (2) How such coherence is developed on complex networks? (3) How to characterize GS on networks? (4) How topology affects GS?
2 Dynamical model and method
The networked oscillators system : & X = F ( X ) − ε ∑ a H ( X − X ),