Generalized synchronization on complex networks Guan Shuguang （管曙光） 16/10/2010

1 Background and motivation 2 Dynamical model and method 3 Development of GS on complex networks 4 Characterizing GS on networks 5 Summary Outlines

1 The background 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. [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). Chaos synchronization

CS GS

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. [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 。 Synchronization on networks

(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? 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. Motivation

2 Dynamical model and method

The auxiliary method [9] H. D. I. Abarbanel et al., PRE 53, 4528 (1996). Drive DResponse R Response R’  R and R’ have the same dynamics, but from different initial conditions  R=R’ (CS) implies R=f (D) (GS)

The auxiliary method on networks Node i Auxiliary node i’

The distance of local GS

The distance of global GS The distance of global CS

(1) 500 non-identical Lorenz oscillators on SFN Color-map of matrix, showing the development of GS is from the hubs to the rest nodes in SF network r randomly in [28, 38] Scale-free network (BA) : ranked by decreasing degree 3 The development of GS on networks

(2) 300 identical Lorenz oscillators on SFN Typical path: STC  GS  CS

The entrainment for nodes 4,104, and 250

SF network  For heterogeneous networks, the hub nodes provide skeleton around which GS is developed. Topology vs local dynamics

SW network Modular network For homogeneous networks, the local dynamics (LLE) determines the development of GS.

Effect of different coupling strategies (a) 300 identical Lorenz oscillators (b) 300 identical Logistic maps SF network

Networked hybrid oscillators SF network Hybrid system: Lorenz oscillators: 95% Rossler oscillators: 5%

4 Characterizing GS on networks The up row ; and the bottom row for 100 identical Lorenz oscillators on SFN. Conditional Lyapunov exponents (CLE): characterizing entrainment of local oscillators

Lyapunov exponents spectra (LES): characterizing global bifurcations SGS regime:  Simple GS (SGS): LE<=0.  Typical path: STC  SGS  CS 100 identical LZ oscillators on SFN.

E.g. 1: SGS state

CGS regime:  Chaotic GS (CGS): at least one LE>0.  Typical path: STC  CGS  CS 300 identical Logistic maps on SF network. (a) (b) (c) (d) CGS

All LEs for 300 identical Logistic maps on SF network.  CGS starts when the first two LEs begin to separate.  the high-d GS manifold gradually collapse to the 3d CS manifold.  the chaotic degree increases.

100 non-identical Lorenz oscillators on SFN, r in [28, 38]. For networked non-identical oscillators:  Coexistence of SGS and CGS.  Typical path: STC  SGS  CGS  Complicated bifurcations: fixed point/limit cycle/CGS.

100 non-identical Lorenz oscillators on SFN, r in [28, 38]. Node 1 is the hub with largest degree; it directly connects node 34, but does not connect node 100. E.g. 1: SGS state

PGS: partial nodes are entrained, others are not. Nodes are entrained from the hubs to others. 100 non-identical Lorenz oscillators on SFN, r in [28, 38]. E.g. 2: PGS state

PGS: nodes are gradually entrained from hub to others. Coexistence of limit cycle and chaos. 100 non-identical Lorenz oscillators on SFN, r in [28, 38]. E.g. 3: Another PGS state

100 non-identical Lorenz oscillators on SFN, r in [28, 38]. E.g. 4: CGS state

Relation between node dynamics? Node i Auxiliary node i’ Entrainment  functional relation among nodes?

Mutual false nearest neighbor (MFNN) method [2] N. Rulkov, et al., PRE 51, 980 (1995).

Direct evidence showing GS relations among node dynamics.

5 Summary GS can occur on many networked oscillator systems. For networked identical oscillator system, a GS regime usually exists before CS. For heterogeneous networks, the hub nodes provide skeleton to develop GS, while for homogeneous networks, the LLE of local dynamics plays a dominant role in GS development. The entrainment of oscillators in networks can be characterized by CLE, and the global bifurcations can be characterized by LES. Direct evidence shows that node dynamics can achieve functional relations though they may not directly connect each other. [10] S. Guan et al., Chaos 19, (2009). [11] S. Guan et al., New J. Phys. 12, (2010).

The End Thank you very much!