平成25年度電気関係学会四国支部連合大会 講演論文集 （2013 徳島大学）
2013 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

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Clustering of Coupled Logistic Map with Bridge
Kenta Ago† , Yoko Uwate† and Yoshifumi Nishio†(† Tokushima University)

1. Introduction
In this study, we focus on “Bridge” in the network.
The bridge is an edge which provide the only route
between two nodes. Here, the edge is line connecting
two nodes. The function of the bridge is called “Bridging function”. In order to analyze complex phenomena of the bridge on the network, we use the coupledmap. Generally, the coupled map is used as generalmodels for complex dynamics, e.g. biological systems,
economic activities and neural networks. Coupled oscillatory systems can also produce interesting phase patterns, e.g. “Clustering”. In this study, we show synchronization phenomena of the network of coupled Logistic map when a coupling strength g and a non-liner
parameter a are changed. Thereby, we observe influenceof the bridge on the network. Moreover, we statisticallyinvestigate synchronized phenomena of the bridge.

2. Network model
We consider the network of coupled logistic map withthe bridge. Network model including the bridge isshown in Fig. 1.

Figure 1: Network model including the bridge.
In Fig. 1, the bridge between 3 and 4 provides the only
route which can flow information or influence from any
contact of 3 to any contact of 4. The number of nodes
and edges are set to 7 and 9. The coupling method of
the logistic map is described as follows.
g ∑xi,n+1 = (1 − g)f (xi,n ) +f (xj,n ),Nj

x1 vs x2 .

x1 vs x3 .

x2 vs x3 .

x3 vs x4 .

x4 vs x5 .

x4 vs x6 .

x4 vs x7 .

x5 vs x6 .

x6 vs x7 .

Figure 2: Simulation result. (a = 4.0, g = 0.35.)
Moreover, we statistically investigate synchronizedphenomena of coupled Logistic map. Here, we define“Asynchronous” by |xi − xj | > 0.1. In this simulation,
we define asynchronous probability from the number of“Corresponding” timing of bursting wave between two
nodes for iteration time n = 100000. Asynchronous
probability of simulation result is shown in Fig. 3. The
parameters are same as Figs. 2 and 3 (b). When g isnear 0, from Fig. 3(a), all maps are higher asynchronousprobability. When g is increase, from Fig. 3(b), thebridge (3-4) has higher asynchronous probability thanother route. Thereby, phase deviation occurs from thebridge statistically. Thus, we could confirm influence ofthe bridge on the network.

j∈Cji= 1, 2, 3..., 7,(1)(a) a = 4.0, g = 0.05.

where x is the fast and slow dynamical variables, gis the coupling strength, Cj is set of nodes which areconnected to node i, and Nj is the number of nodes
which are connected to node i. We apply Logistic map
to the function f (). Logistic map is described as follows.
xn+1 = axn (1 − xn ),(2)

where a is the non-liner parameter.

3. Simulation result
First, when a coupling strength g and a non-liner parameter a are changed, we show synchronization of thenetwork of Logistic map which produce chaotic behavior. One of the simulation result is shown in Fig. 2.
Here, two parameters are a = 4.0 and g = 0.35. In thissimulation, iteration time is n = 100000. From Fig. 2,synchronization of bridge (3-4) is weak compared withother. Therefore, the bridge has a strong influence onnetwork. Also, 1-2 and no-coupled 5-7 are full synchronization. Because these aﬀected from same nodes.

(b) a = 4.0, g = 0.35.

Figure 3: Asynchronous probability.

4. Conclusions
In this study, we have studied influence of the bridgein the network via synchronization phenomena of coupled Logistic map. From simulation results, in the network with a bridge, clustering of maps is occurred froma bridge. Moreover, we statistically confirmed that thebridge has a strong influence on the network.

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